applied sciences

Article

Flow Regime, Slug Frequency and Wavelet Analysisof Air/Newtonian and Air/non-NewtonianTwo-Phase Flow

Munzarin Morshed 1, Muhammad Saad Khan 2 , Mohammad Azizur Rahman 2,*and Syed Imtiaz 1

1 Department of Process Engineering, Memorial University of Newfoundland, St. John’s, NL A1C 5S7,Canada; mm3112@mun.ca (M.M.); simtiaz@mun.ca (S.I.)

2 Department of Petroleum Engineering, Texas A&M University at Qatar, Doha 23874, Qatar;khansaaad@gmail.com

* Correspondence: marahman@tamu.edu; Tel.: +974-4423-0601

Received: 7 April 2020; Accepted: 4 May 2020; Published: 8 May 2020�����������������

Featured Application: Application of Newtonian and non-Newtonian fluids in horizontal flowloop conditions for directional drilling operations.

Abstract: This study focused on gas/Newtonian and gas/non-Newtonian two-phase horizontalfluid flow behavior by analyzing their flow regime identification and flow structural analysis on ahorizontal flow loop apparatus. This involved the recognition of two-phase flow regimes for this flowloop and validation with existing flow maps in the literature. In addition, the study included flowpattern identification via wavelet analysis for gas/Newtonian and gas/non-Newtonian two-phasefluid flow in a horizontal flow loop apparatus. Furthermore, the study was extended to the detailedexamination of slug frequency in the presence of air/Newtonian and air/non-Newtonian fluid flow,and the predicted slug frequency model was applied to the studied systems. The obtained resultssuggest that the flow regime maps and slug frequency analysis have a significant impact. The obtainedpressure sensor results indicate that the experimental setup could not provide high-frequency andhigh-resolution data; nevertheless, wavelet decomposition and wavelet norm entropy were calculated.It offered recognizable flow characteristics for bubble, bubble-elongated bubble, and slug flow patterns.Therefore, this study can provide deep insight into intricate multiphase flow patterns, and the waveletcould potentially be applied for flow analysis in oil and gas pipelines.

Keywords: air/Newtonian and air/non-Newtonian fluid; slug frequency; two-phase flow regimes;superficial gas velocity; wavelet analysis

1. Introduction

Multiphase flows are considered as complicated flow phenomena compared to a single flow [1].There are essential features of multiphase flows whose modeling outcome is contentious and structuralexplanation is still unexplored. The most common type of multiphase flow in almost all chemical,petroleum, and production industries is the two-phase gas/liquid flow [2,3]. Different forms of flowpatterns may be observed when two or more phases flow simultaneously [2]. In pipe cross-sections,unpredictable turbulent flow structures generating highly asymmetric volume distribution are achallenge in experiential investigations [4,5]. This kind of unstable flow condition complicates themeasurement process and may create difficulties in capturing actual flow conditions. There are alsoinstances where the existing theoretical solution or experimental results cannot describe specificphysical properties [1,6].

Appl. Sci. 2020, 10, 3272; doi:10.3390/app10093272 www.mdpi.com/journal/applsci

Appl. Sci. 2020, 10, 3272 2 of 27

Flow regimes or patterns are amongst the most significant aspects of multiphase flow.The structural flow distribution of different phases in a pipe is known as the flow pattern orflow regime. The flow regime depends on the inertia force, buoyancy force, flow turbulence,and surface tension, which are altered by fluid properties, flow rates, pipe diameter, and pipepredilection [7,8]. Different forms of flow patterns may be observed when two phases of gas/Newtonianand gas/non-Newtonian flow simultaneously. Some of the conventional distributions are stratifiedflow, where the liquid and gas phase is separated, and the gas flows on top as it is lighter than theliquid; bubbly flow, where there is a dispersion of small-sized bubbles within the liquid; slug flow, inwhich each gas bubble forms a large slug shape (often a bullet shape); and annular flow where liquidflows as a film on the wall of the pipe. For gas/Newtonian and gas/non-Newtonian flow, several flowmaps exist to predict the flow patterns. The Taitel and Dukler [7] and Mandhane et al. [9] flow mapsfor gas/Newtonian flow, and the Chhabra and Richardson [10] flow map for gas/non-Newtonian floware the most frequently used.

Among the flow regimes, slug flow is the most frequent two-phase flow phenomena experiencedin horizontal or near horizontal pipelines in practical applications [11–13]. The presence of slug flowsin a pipeline can trigger a great range of design and operational problems, including the exertionof kinetic force on fittings and fittings, pressure cycling, control instability, and inadequate phaseseparation. If the flow rate is raised, much of the liquid is swept out, and the liquid handling capacityof the receiving facilities may be overwhelmed. In the case of a slug greater than the slow catchercapacity, the facilities can be flooded and destroyed. Therefore, multiple operational problems, such aspipeline network instability and damage to equipment due to high-pressure fluctuation or vibrationof the system, can be caused by slug flow [14,15]. This can also be termed a water hammering effect.Therefore, slug flow and slug frequency analysis has been one of the primary research interests inmultiphase flow.

To measure the hydrodynamic behavior of multiphase flow, it is essential to understand flowpattern under specific flow conditions [2,16]. Two-phase flow implies gas and liquid flow through apipeline system, simultaneously. The gas and liquid interface is deformable, so it is hard to predict theregion occupied by a gas or liquid phase [17]. When two phases flow through a pipeline, differenttypes of interfacial distribution can form. The variety of flow patterns mostly depends upon the inputflux of the two phases, the size and assembly of the pipe, the physical properties of the fluid, etc. Manyexperimental studies on gas/Newtonian or solid/Newtonian fluid flow have been conducted [18–24].Nevertheless, a limited number of studies have observed non-Newtonian multiphase flow [25,26].

Xanthan gum is the most commonly used industrial biopolymer for thickening and stabilizingan aqueous system. Xanthan gum solution has significant thermal stability with pseudoplasticproperties [27]. Due to these properties, it has a substantial application in petroleum industries. In oilindustries, xanthan gum is widely used in drilling fluid [28]. Xanthan gum is also broadly used in thefood industry, cosmetics, and pharmacological products [27,29].

Tutu [30] and Drahos et al. [31] characterized two-phase horizontal flow regime pressure fluctuation.Drahos et al. [31] used a probability density function (PDF), where a strain gauge pressure transducerwas used in a 50 mm inner diameter (ID) Perspex pipe. Sun et al. [32] used norm entropy waveletdecomposition to analyze gas/liquid two-phase flow pressure signal data across a bluff body. Here,the inner pipe diameter was 50 mm, and, from a piezoresistive differential pressure sensor, the pressuresignals were analyzed using four levels and four scales of Daubechies wavelet (db4), which provided 16wavelet packet coefficients. This study [32] also suggested an entropy-based two-phase flow map withan identification rate of 95%. Blaney [33] used gamma rays to identify flow regimes, and continuouswavelet transforms to analyze gamma count data. Park and Kim [34] applied the wavelet packettransform to analyze pressure fluctuations in a vertical bubble column.

Furthermore, De Fang et al. [35] also used wavelet analysis to understand the gravity differentialpressure fluctuation signal perpendicular to the horizontal flow of different flow patterns and the flowpattern transition of gas/liquid two-phase flow in the horizontal pipe. Here, the Haar wavelet with

Appl. Sci. 2020, 10, 3272 3 of 27

six levels was used to decompose the pressure signal, and the energy value was then obtained foreach scale. For identifying a two-phase flow regime, Elperin and Klochko [36] also used an eight-leveldb4 wavelet transformation to process time series of the measured differential pressure fluctuation.In this study, using pressure transducer signal data of different flows, the pattern was analyzed usingwavelet transformation to determine the pressure signal characteristics of various flow regimes.

To identify the two-phase flow regime mapping and avoid slug flow in horizontal pipelines,a clear understanding of the Newtonian and non-Newtonian fluid flow is necessary for an accurateand inherently safer design for any flow system. The uniqueness of this study is that the experimentswere performed in a setup with 73.66 mm ID and approximately 19 m flow loop in the presence ofgas/Newtonian and gas/non-Newtonian two-phase horizontal flow. This flow loop has a horizontal,vertical, and inclined test section connected; however, the study is only on horizontal flow aspects.Another primary focus of this study is to understand the characteristics of slug frequency analysisfor both air/Newtonian and air/non-Newtonian flow conditions in the presence of Xanthan gum asa non-Newtonian fluid. This study also recognizes the characteristics of wavelet analysis for bothair/Newtonian and air/non-Newtonian flow conditions, i.e., water and Xanthan gum.

2. Materials and Methods

2.1. Materials

Water is used as a Newtonian fluid in this study. For non-Newtonian fluid experiments, a 0.1 wt%solution of biopolymer, i.e., Xanthan gum, was used (obtained from Merck, Germany). To conducttwo-phase experiments, compressed air was used for gas/Newtonian fluid and gas/non-Newtonianexperiments. The air was compressed via an in-house compression facility.

2.2. Experimental Methods

The experiments were performed in a flow loop system that has a horizontal, vertical, and inclinedsection. However, in this paper, we are only considering the 4-m horizontal section as our test section.The experimental setup is a 60-m-long closed cycle system for water and open cycle system for air. Theliquid is pumped by a 5 HP pump that creates the required large volume water flow through DN80 or2.9 ID PVC clear pipes. The airline of the flow loop comprises of DN15 and DN25 mild steel pipes,which supply air from the lab air supply at 670 kPa (100 psi) shut-in pressure. It also includes a DN 25ball check valve just before the air and the liquid mixing zone to prevent any liquid from entering theair pipeline. There are two Omega PX603-100G5V pressure transducers with a range of 0 to 100 psiin the 2-m long horizontal test section. There were some specific experimental conditions used forthis setup. The airflow range was approximately 85 L/min to 3300 L/min, and the water flow rangedfrom almost 250 L/min to 850 L/min. Within this range, the experimental setup mostly provides slugflow for two-phase flow, as well as providing bubble flow and wavy flow at some intervals. Figure 1presents a schematic representation of the experimental setup. For this study, both gas/Newtonianand gas/non-Newtonian fluid flow cases were considered. A Universal Data Acquisition System fromNational Instruments was used to collect all types of data from the flowmeter and sensors, whichconsisted of four NI 9219 universal modules with four channels, each providing 100 samples persecond. The experimental conditions are reported in Table 1, including brief details of the rheologicalmodel used for xanthan gum solutions.

Table 1. The chemicals and experimental parameters used in this study.

Newtonian Fluid Water

Non-Newtonian Fluid 0.1% xanthan gum solution (Power Law Index, n = 0.81) and (Power Law Index,m or k = 0.009344)

Liquid velocity range 1.5 to 2.5 m/sAir velocity 2.9 to 6.4 m/s

Appl. Sci. 2020, 10, 3272 4 of 27

Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 31

Figure 1. Schematic of the experimental setup (horizontal test section) with a data acquisition system.

Table 1. The chemicals and experimental parameters used in this study.

Newtonian Fluid Water

Non-Newtonian

Fluid

0.1% xanthan gum solution (Power Law Index, n

= 0.81) and (Power Law Index, m or k = 0.009344)

Liquid velocity

range 1.5 to 2.5 m/s

Air velocity 2.9 to 6.4 m/s

Slug flow is the most frequent two-phase flow phenomenon experienced in horizontal or near horizontal pipelines in practical applications. Slug flow in pipes is encountered in different fields, such as the production and transportation of oil and gas, food industry, chemical industry, and other industrial applications. Slug frequency, in addition to water hammering, leads to various operational problems, such as pipeline network instability, equipment damage, pressure fluctuations, and vibration of the system. In the oil and gas production industries, slug flow also influences the internal corrosion rate increase of carbon steel pipelines. Slug flow creates high turbulence, which can break the pipe wall inhibitor’s protection layer [37].

2.2.1. Slug Velocity

In 1965 and 1977, Hubbard [38, 39] and Otten & Fayed [40], respectively, experimentally measured the slug velocity by observing a particular slug movement in test sections. They both obtained a relationship between slug velocity and no-slip mixture velocity by plotting the experimentally measured slug velocity against the no-slip mixture velocity. The slug flow model of

Figure 1. Schematic of the experimental setup (horizontal test section) with a data acquisition system.

Slug flow is the most frequent two-phase flow phenomenon experienced in horizontal or nearhorizontal pipelines in practical applications. Slug flow in pipes is encountered in different fields,such as the production and transportation of oil and gas, food industry, chemical industry, and otherindustrial applications. Slug frequency, in addition to water hammering, leads to various operationalproblems, such as pipeline network instability, equipment damage, pressure fluctuations, and vibrationof the system. In the oil and gas production industries, slug flow also influences the internal corrosionrate increase of carbon steel pipelines. Slug flow creates high turbulence, which can break the pipewall inhibitor’s protection layer [37].

2.2.1. Slug Velocity

In 1965 and 1977, Hubbard [38,39] and Otten & Fayed [40], respectively, experimentally measuredthe slug velocity by observing a particular slug movement in test sections. They both obtaineda relationship between slug velocity and no-slip mixture velocity by plotting the experimentallymeasured slug velocity against the no-slip mixture velocity. The slug flow model of Hubbard [39]showed better agreement at higher slug velocity. Hubbard [39] described the relationship as:

vs = 1.25 vm (1)

Hubbard & Dukler [39] also predicted the actual average gas velocity as in Equation (2), whichalso agreed with other experimental data [41].

vG = 1.19 vm (2)

Appl. Sci. 2020, 10, 3272 5 of 27

It is assumed that the Hubbard & Dukler [39] slug flow model was verified based on one dominantpresupposition that the liquid slug velocity and the maximum gas phase velocity should be similar.Therefore, theoretically, the no-slip mixture velocity should be equal to the slug velocity:

vs

vm= C (3)

where C is a constant. Theoretically, C is assumed to be 1.0 for air–water two-phase flow. Hubbard [38],Rosehart et al. [42], and Gregory & Scott [41] considered C values of 1.25, 1.26, and 1.35, respectivelyfor air–water flow. These C values may have varied because of different experimental setups andconditions [40]. For non-Newtonian/air two-phase flow, Otten & Fayed [40] compared their results withthose of Rosehart et al. [42]. The obtained results suggested that the air/Carbopol 941 concentrationreported by Otten & Fayed [40] increased from 0.75% to 0.2% and C values increased from 1.36 to 1.41,whereas for the same concentration, the C values of Rosehart et al. (1975) varied from 1.54 to 1.98.

2.2.2. Slug Frequency

Different correlations can predict slug frequency. However, Gregory & Scott [41] suggested atrustworthy velocity-dependent empirical equation where slug frequency was correlated with a formof Froude number, which is described as:

N fs =vlgd

(v0

m

)2

vm+ vm

(4)

Here, v0m was taken as 6 m/s and from the graphs of the slug frequency vs. slug Froude number by

Gregory & Scott [41], the following equation was derived:

fs = 0.0157(N fr

)1.20

slugsec−1 . (5)

From Equation (5), Gregory & Scott [41] described a slug frequency correlation based on theirliquid-gas two-phase flow experimental data where water and carbon dioxide are used in a 19-mmID pipe:

fs = 0.0226[ vs

lgd

(19.75vm

+ vm

)]1.2

(6)

where vm and vsl represent the mixture velocity and superficial liquid velocity of liquid and gas,

respectively. Therefore, this slug frequency can be combined with the Froude number established forsuperficial fluid velocity.

Greskovich & Shrier [43] reorganized the Gregory & Scott [41] correlation as follows:

fs =[0.0425

vsl

vm

(2.02

d+

v2m

gd

)] 65

(7)

Later, Zabaras & others [44] described another correlation based on 399 data points with thesmallest average absolute error and standard deviation for both horizontal and inclined pipe flow.This correlation is a modification of the Gregory & Scott [41] correlation, using the imperial unit, whichis shown in Equation (8), where θ is the inclination angle. The experiment was conducted in a 2.54 cmand 10.16 cm ID pipe with air and water.

fs =[0.0425

vsl

gd

( 10.0506vm

+ vm

)] 65 [

0.836 + 2.7sin0.25θ]

(8)

Appl. Sci. 2020, 10, 3272 6 of 27

Heywood & Richardson [45] determined the liquid volume fraction for air–water two-phase flowutilizing the gamma-ray technique in a 4.191 cm ID horizontal pipe. To determine the liquid volumefraction, they used a power spectral density function and probability density function. These featuresare also helpful to decide on different slug flow characteristics, such as the value of the average film andslug volume fraction, average slug frequency, and average slug length. The slug frequency correlationwas determined by curve fitting the data, resulting in Equation (9), in which λ is the liquid volumefraction, where λ = vs

l /(vs

l + vsg

), and d is the pipe diameter.

fs =[0.0462 λ

(1

0.0126d+

v2m

gd

)]1.02

(9)

Shea et al. [46] developed a correlation describing slug frequency as a function of pipe length.In Equation (10), vS

l is the superficial liquid velocity, d is the pipe diameter, and lp is the pipe length.This correlation is based on the curve fitting of field and laboratory data, and is not based ontheoretical analysis. In this equation, it is also shown that the slug frequency is inversely dependent onthe pipe length lp, which does not agree with the other theoretical analysis. According to Al-Safran [14],the OLGA 2000 slug tracking model had some time delay problem between two slugs; to solve thisissue, the Shea et al. [46] correlation was initially used. Moreover, pipe length can be questionable for along-distance transmission system with hilly conditions:

fs = 0.47

(

vsl

)1.5

l1.1p d2.4

0.5

(10)

Picchi et al. [47] described a slug frequency equation that considers the rheology of theshear-thinning fluid. This equation is a modified version of the Gregory & Scott [41] correlation.

In Equation (11), Rew =ρwvs

l duw

is the water Reynolds number and Ren =dnvlnρn

m8n−1( 1+3n4n )

n−2 is the power-law

fluid Reynolds number under superficial conditions, where n and m are fluid behavior indexes.

fs = 0.0448[

vlngd

(32.2014

vmn

+ vmn

)]0.88

n−2.85( Ren

Rew

)0.07(11)

2.2.3. Wavelet Analysis

Wavelet analysis is an effective means of signal processing. Wavelets are uneven and asymmetricalwaveforms of adequately limited duration with an average value of zero. Wavelet analysis breaks upthe mother wavelet signal into a shifted and scaled version. The typical wavelet transformation of asine wave is presented in Figure 2.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 31

= 0.0425 10.0506 + 0.836 + 2.7 . (8)

Heywood & Richardson [45] determined the liquid volume fraction for air–water two-phase flow utilizing the gamma-ray technique in a 4.191 cm ID horizontal pipe. To determine the liquid volume fraction, they used a power spectral density function and probability density function. These features are also helpful to decide on different slug flow characteristics, such as the value of the average film and slug volume fraction, average slug frequency, and average slug length. The slug frequency correlation was determined by curve fitting the data, resulting in Equation (9), in which is the liquid volume fraction, where = ( + )⁄ , and d is the pipe diameter.

= 0.0462 10.0126 + . (9)

Shea et al. [46] developed a correlation describing slug frequency as a function of pipe length. In Equation (10), is the superficial liquid velocity, d is the pipe diameter, and is the pipe length. This correlation is based on the curve fitting of field and laboratory data, and is not based on theoretical analysis. In this equation, it is also shown that the slug frequency is inversely dependent on the pipe length lp, which does not agree with the other theoretical analysis. According to Al-Safran [14], the OLGA 2000 slug tracking model had some time delay problem between two slugs; to solve this issue, the Shea et al. [46] correlation was initially used. Moreover, pipe length can be questionable for a long-distance transmission system with hilly conditions:

= 0.47 ( ) . . . . (10)

Picchi et al. [47] described a slug frequency equation that considers the rheology of the shear-thinning fluid. This equation is a modified version of the Gregory & Scott [41] correlation. In Equation (11), = is the water Reynolds number and = is the power-law fluid

Reynolds number under superficial conditions, where n and m are fluid behavior indexes.

= 0.0448 32.2014 + . . . (11)

2.2.3. Wavelet Analysis

Wavelet analysis is an effective means of signal processing. Wavelets are uneven and asymmetrical waveforms of adequately limited duration with an average value of zero. Wavelet analysis breaks up the mother wavelet signal into a shifted and scaled version. The typical wavelet transformation of a sine wave is presented in Figure 2.

Figure 2. Wavelet transformation of a sine wave.

In Fourier analysis, the signals are decomposed into different sine waves. Therefore, an irregular wavelet performs better than the steady sine wave for rapidly changing signals as it can provide

Figure 2. Wavelet transformation of a sine wave.

Appl. Sci. 2020, 10, 3272 7 of 27

In Fourier analysis, the signals are decomposed into different sine waves. Therefore, an irregularwavelet performs better than the steady sine wave for rapidly changing signals as it can providebetter information about specific and appropriate locations. Wavelet analysis can also show anydiscontinuity, breakdown, trend, noise, coefficient, etc., of signals [48]. In this study, wavelet analysiswas conducted using the MATLAB toolbox. There are two types of wavelet analyses, namely, discretewavelet transform and continuous wavelet transform. There are various subgroups of these two typesof wavelet transforms.

Contentious Wavelet Transform (CWT)

The continuous wavelet transforms (CWT) is a function of the shifted and scaled version of thewavelet function Ψ multiplied by the summation of the overall time of the signal. However, scalingmeans compressing or stretching the wavelets. A scale factor is used to represent the scaling, wherethe wavelet is more compressed when the scale factor is smaller. Wavelet sifting means hastening ordetaining its onset.

C(position, scale) =∫∞

−∞

f (x)Ψ(position, scale, x)dt (12)

where C is the wavelet coefficient of the CWT as a function of position and scale [48].

Discrete Wavelet Transform (DWT)

The discrete wavelet transform is a wavelet transform where the wavelets are separately sampled.In this analysis, the original signal is divided into two parts, namely, approximations and details.The estimate a is the low pass filter where the low-frequency components of the original signal areseparated, and the detail d is the high pass filter where high-frequency components pass. Moreover, theoriginal signal x is not only isolated in one level, but the approximation a is also decomposed into muchlower level (k = 3) components, known as multiple level decomposition, which is shown in Figure 3.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 31

better information about specific and appropriate locations. Wavelet analysis can also show any discontinuity, breakdown, trend, noise, coefficient, etc., of signals [48]. In this study, wavelet analysis was conducted using the MATLAB toolbox. There are two types of wavelet analyses, namely, discrete wavelet transform and continuous wavelet transform. There are various subgroups of these two types of wavelet transforms.

Contentious Wavelet Transform (CWT)

The continuous wavelet transforms (CWT) is a function of the shifted and scaled version of the wavelet function multiplied by the summation of the overall time of the signal. However, scaling means compressing or stretching the wavelets. A scale factor is used to represent the scaling, where the wavelet is more compressed when the scale factor is smaller. Wavelet sifting means hastening or detaining its onset. ( , ) = ( ) ( , , ) (12)

where C is the wavelet coefficient of the CWT as a function of position and scale [48].

Discrete Wavelet Transform (DWT)

The discrete wavelet transform is a wavelet transform where the wavelets are separately sampled. In this analysis, the original signal is divided into two parts, namely, approximations and details. The estimate a is the low pass filter where the low-frequency components of the original signal are separated, and the detail d is the high pass filter where high-frequency components pass. Moreover, the original signal x is not only isolated in one level, but the approximation a is also decomposed into much lower level (k = 3) components, known as multiple level decomposition, which is shown in Figure 3.

The significant difference between CWT and DWT is that CWT operates at every scale up to the maximum value, whereas, in DWT, the range and positions can be preselected and, in that way, the size of the analysis reduces its size and is thereby faster and more precise.

= + = + + = + + +

Figure 3. Multiple level discrete wavelet analysis.

Mathematically, for j scales and k levels, the approximate information ( ) can be the summation of estimated coefficients , and the scale function , ( ), as shown in Equation (13). Similarly, the detail information ( ) can also be described as the summation of approximate coefficients , and the scale function ( ) in Equation (14): ( ) = , , ( ) (13)

( ) = , ( ) (14)

Figure 3. Multiple level discrete wavelet analysis.

The significant difference between CWT and DWT is that CWT operates at every scale up to themaximum value, whereas, in DWT, the range and positions can be preselected and, in that way, thesize of the analysis reduces its size and is thereby faster and more precise.

Mathematically, for j scales and k levels, the approximate information f ja (x) can be the summation

of estimated coefficients a j,k and the scale function ϕ j,k(x), as shown in Equation (13). Similarly,

the detail information f jd(x) can also be described as the summation of approximate coefficients d j,k

and the scale function Ψ jk(x) in Equation (14):

f ja (x) =

∑k

a j,kϕ j,k(x) (13)

Appl. Sci. 2020, 10, 3272 8 of 27

f jd(x) =

∑k

d j,kΨ jk(x) (14)

A standard means of applying this is via logarithmic discretization of the scale s and thenconnecting it to the step size. The step size is the values between the translation parameter τ.The equation is adapted from Gao & Yan (2010) [49] and shown below, where Z is an integer parameter:

{s=s j

0

τ=kτ0s j0

τ0 , 0; s0 < 1 ( jεZ, kεZ) (15)

Ψ jk(x) = s−0.5 j0 Ψ(

x

s j0

− kτ0) (16)

Ψ jk(x) = 2−0.5 jΨ(x2 j − k) (17)

where j is the scale and k is the level of the wavelet. Equation (18) is the base wavelet equation.Addison (2017) [50] assumed s0 = 2 and τ0 = 1; therefore, Equation (19) can be derived and, finally,the discrete wavelet transform is obtained:

W( j, k) =[

f (x), Ψ jk(x)]= 2−0.5 j

∫∞

−∞

f (x) Ψ( x

2 j − k)dx (18)

f (x) =∑j,k

C j,kΨ jk(x) (19)

In Equation (18), f (x) is the original signal and, in Equation (19), C j,k is the wavelet coefficient.For multilevel wavelet analysis, there are many types of orthogonal wavelet transformations, whichdetermine the shape of the wavelet. Among them, the Daubechies wavelet is one of the most commonorthogonal wavelet transformations.

Daubechies Wavelet

The Daubechies wavelet uses scalar products with scaling wavelets and signals to calculate movingaverage and difference. This method allows for obtaining a good range of signal data to compute themedian and deviation. Daub4 is the most accepted and straightforward way of analyzing wavelets.If we consider a signal x constituting n number of values, then the daub4 transformation creates the

mapping xDk→ (ak

|dk) to its approximation ak and details dk sub-signals for k levels:

am = x.Ukm (20)

dm = x.Ψkm (21)

where each value of am and dm are the scaler products. Ukm is the scaling signal and Ψk

m is the waveletat level k [51].

Wavelet Packet Analysis

In DWT, the primary signal is decomposed into an approximation and details, and theapproximation is divided into a second-level approximation and features; thus, n levels of decompositioncan be undertaken. In wavelet packet analysis, both the details and the estimate can be decomposed,which means the signal can be encoded in 2n ways. The wavelet packet decomposition tree is shownin Figure 4.

Appl. Sci. 2020, 10, 3272 9 of 27Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 31

Figure 4. Wavelet packet analysis decomposition tree.

In the MATLAB toolbox, the entropy-based criterion is used to find the most desirable wavelet decomposition. Wavelet packet transformation provides many bases, and the best tree base can be found by the entropy criterion [48]. Wavelet packets are the general form of orthogonal wavelets. This splits detailed spaces to yield more prime decomposition. Coifman & Wickerhauser (1992) [52] explained the wavelet packet transformation equation as the following: ( ) = √2 ℎ (2 − )

( ) = √2 (2 − ); = 0,1,2, … = 0,1, … (22)

where the two filters hk and gk are associated with the scaling function ( ) and the base wavelet function ( ) [49].

Wavelet Entropy

Wavelet entropy represents the non-uniformity of states, which is an ideal parameter for measuring the ordering of unsteady signals [53]. It can also provide information about the dynamic process and the signal potential. When the coefficient matrix of the wavelet transformation is represented by a probability distribution, the calculated wavelet entropy represents the randomness of the pattern [54]. The wavelet packet decomposition is an orthogonal function, which means the total energy entropy of the original signal should be the summation of the coefficient energy entropy [32]. The wavelet entropy energy can be defined as in Equation (23), where = / ∑ is the percentage of coefficient energy of the original signal [55]:

= − (23)

In this study, the norm entropy was used to analyze the pressure signal. In an orthonormal basis entropy, s is the signal, is the coefficient of s, and E is the entropy function such that (0) = 0 and ( ) = ∑ ( ). This entropy formula was used in MATLAB to calculate norm entropy. The concentration in is a norm where 1 ≤ < 2. Now, ( ) = | | so ( ) = ∑ | | = | | for the norm entropy method [48]. The wavelet entropy can find small or abnormal frequencies, and wavelet entropy can thus find different characteristics of multiphase flow. Therefore, this study aims to characterize a two-phase flow pattern using norm entropy based on wavelet packet decomposition of the pressure signal. This method followed the steps shown in Figure 5.

x

a1

aa2

aaa3 daa3

da2

ada3 dda3

d1

ad1

aad3 dad3

dd1

add3 ddd3

Figure 4. Wavelet packet analysis decomposition tree.

In the MATLAB toolbox, the entropy-based criterion is used to find the most desirable waveletdecomposition. Wavelet packet transformation provides many bases, and the best tree base can befound by the entropy criterion [48]. Wavelet packets are the general form of orthogonal wavelets.This splits detailed spaces to yield more prime decomposition. Coifman & Wickerhauser (1992) [52]explained the wavelet packet transformation equation as the following:

v j2i(x) =

√2∑k

hkv ji (2x− k)

v j2i+1(x) =

√2∑k

gkv ji (2x− k)

; i = 0, 1, 2, . . . and k = 0, 1, . . .m (22)

where the two filters hk and gk are associated with the scaling function ϕ jk(x) and the base waveletfunction Ψ jk(x) [49].

Wavelet Entropy

Wavelet entropy represents the non-uniformity of states, which is an ideal parameter for measuringthe ordering of unsteady signals [53]. It can also provide information about the dynamic process andthe signal potential. When the coefficient matrix of the wavelet transformation is represented by aprobability distribution, the calculated wavelet entropy represents the randomness of the pattern [54].The wavelet packet decomposition is an orthogonal function, which means the total energy entropy ofthe original signal should be the summation of the coefficient energy entropy [32]. The wavelet entropyenergy can be defined as in Equation (23), where Pi = Ei/

∑ni=1 Ei is the percentage of coefficient energy

of the original signal [55]:

EN = −n∑

i=1

PilogPi (23)

In this study, the norm entropy was used to analyze the pressure signal. In an orthonormalbasis entropy, s is the signal, si j is the coefficient of s, and E is the entropy function such thatE(0) = 0 and E(s) =

∑i E(si). This entropy formula was used in MATLAB to calculate norm entropy.

The concentration in lP is a norm where 1 ≤ P < 2. Now, E(s) = |si|P so E(s) =

∑i|s|

P = ||s||PP forthe norm entropy method [48]. The wavelet entropy can find small or abnormal frequencies, andwavelet entropy can thus find different characteristics of multiphase flow. Therefore, this study aims tocharacterize a two-phase flow pattern using norm entropy based on wavelet packet decomposition ofthe pressure signal. This method followed the steps shown in Figure 5.

Wavelet Packet Analysis of the Experimental Data

In this study, a pressure transducer provided time-domain pressure fluctuations, which wereanalyzed using wavelet packet analysis. As mentioned previously, the existing experimental setuponly provides slug flow, and the dispersed bubble flow regime and pressure signal also show specificcharacteristics for each kind of flow regime. The data acquisition system collected pressure transducer

Appl. Sci. 2020, 10, 3272 10 of 27

signals with a sampling frequency of 100 Hz. Overall, 10,000 data points were used in 1-D waveletanalysis using MATLAB (details can be found in the above sections). The MATLAB processed datawere taken into account for further analysis.Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 31

Figure 5. The steps of wavelet decomposition for different flow pattern identification.

Wavelet Packet Analysis of the Experimental Data

In this study, a pressure transducer provided time-domain pressure fluctuations, which were analyzed using wavelet packet analysis. As mentioned previously, the existing experimental setup only provides slug flow, and the dispersed bubble flow regime and pressure signal also show specific characteristics for each kind of flow regime. The data acquisition system collected pressure transducer signals with a sampling frequency of 100 Hz. Overall, 10,000 data points were used in 1-D wavelet analysis using MATLAB (details can be found in the above sections). The MATLAB processed data were taken into account for further analysis.

3. Results and Discussions

3.1. Flow Regime Mapping for Horizontal Flow

3.1.1. Air/Newtonian Flow Map

The experimental values were used to derive a flow regime map for horizontal pipe flow. This flow regime map was compared with that of Taitel & Dukler [7] and Mandhane et al. [9], where water and superficial air velocity was used. In the Taitel & Dukler [7] flow map for a horizontal pipe in Figure 6, most of the experimental data points fall in the respective flow regime area. However, the Taitel & Dukler [7] flow map better predicted the dispersed bubble flow for high gas/water velocity than the Mandhane et al. [9] flow map for this experimental setup.

Pressure Fluctuation Signal

1-D Wavelet Packet

Decomposition

Wavelet Spectrum for Differnt of Decomposition

Norm Entropy Analysis

Plot data

Identify the flow Pattern

Develop the Flow Map Based on Norm Entropy

Figure 5. The steps of wavelet decomposition for different flow pattern identification.

3. Results and Discussions

3.1. Flow Regime Mapping for Horizontal Flow

3.1.1. Air/Newtonian Flow Map

The experimental values were used to derive a flow regime map for horizontal pipe flow. This flowregime map was compared with that of Taitel & Dukler [7] and Mandhane et al. [9], where water andsuperficial air velocity was used. In the Taitel & Dukler [7] flow map for a horizontal pipe in Figure 6,most of the experimental data points fall in the respective flow regime area. However, the Taitel &Dukler [7] flow map better predicted the dispersed bubble flow for high gas/water velocity than theMandhane et al. [9] flow map for this experimental setup.

In Figure 7, the Mandhane et al. [9] flow map is provided where the data for the slug and dispersedbubble flow data were fitted in the graph accordingly. The map can predict the slug and bubbleflow regimes. Nevertheless, for high gas and water flow rates, this map cannot predict the dispersedbubble flow regime due to experimental conditions and the pipe diameter.

Appl. Sci. 2020, 10, 3272 11 of 27Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 28

Figure 6: Comparison of the Taitel & Dukler [7] (adapted) flow map with experimental data for horizontal gas/Newtonian flow.

In Figure 7, the Mandhane et al. [9] flow map is provided where the data for the slug and dispersed bubble flow data were fitted in the graph accordingly. The map can predict the slug and bubble flow regimes. Nevertheless, for high gas and water flow rates, this map cannot predict the dispersed bubble flow regime due to experimental conditions and the pipe diameter.

Figure 7: Comparison of the Mandhane et al. [9] (adapted) flow regime map with experimental data obtained for horizontal gas/Newtonian flow.

3.1.2. Air/non-Newtonian flow map Researchers have also developed different flow pattern maps for horizontal, vertical, and

inclined gas/non-Newtonian flow. For horizontal gas/non-Newtonian fluid, Chhabra & Richardson [56] developed a flow pattern map by slightly modifying the Mandhane et al. [9] horizontal flow pattern map (Figure 8). This map was developed for evaluating the literature and verified using 3700 data points of gas/non-Newtonian shear-thinning liquid mixture flow with 70% certainty. However,

0.001

0.01

0.1

1

10

0.01 0.10 1.00 10.00 100.00

Liqu

id (w

ater

) Sup

erfit

ial V

eloc

ity, vls

m/s

Gas Superfitial Velocity, vgs m/s

Slug

Dispersed Bubble

Dispersed Bubble

Elongated Buuble / PlugSlug

Anular

WavyStatified

0.001

0.01

0.1

1

10

0.01 0.10 1.00 10.00 100.00

Supe

rfiti

al L

iqui

d (w

ater

) Vel

ocity

, vls

m/s

Superfitial Gas Velocity, vgs m/s

SlugDispersed Bubble

Dispersed Bubble

Elongated Buuble / Plug Slug

Anular

WavyStatified

Figure 6. Comparison of the Taitel & Dukler [7] (adapted) flow map with experimental data forhorizontal gas/Newtonian flow. Copyright AIChE Journal, 2020.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 29

Figure 6: Comparison of the Taitel & Dukler [7] (adapted) flow map with experimental data for horizontal gas/Newtonian flow.

In Figure 7Figure 7 below, the Mandhane et al. [9] flow map has been is provided where the data 320 for the slug and dispersed bubble flow data were fitted in the graph accordingly. The map can predict 321 the slug and bubble flow regimes. Nevertheless, for high gas and water flow rates, this map cannot 322 predict the dispersed bubble flow regime due to experimental conditions and the pipe diameter. 323

324

Figure 7: Comparison of the Mandhane et al. [9] (adapted) flow regime map with experimental data obtained for horizontal gas/Newtonian flow.

3.1.2. Air/non-Newtonian flow map 325 Researchers have also developed different flow pattern maps for horizontal, vertical, and 326

inclined gas/non-Newtonian flow. In Figure 8 fFor horizontal gas/non-Newtonian fluid, Chhabra & 327 Richardson [56] developed a flow pattern map by slightly modifying the Mandhane et al. [9] 328 horizontal flow pattern map (Figure 8Figure 8). This map has been was developed for evaluating the 329 literature and verified using 3700 data points of gas/non-Newtonian shear-thinning liquid mixture 330

0.001

0.01

0.1

1

10

0.01 0.10 1.00 10.00 100.00

Liqu

id (w

ater

) Sup

erfit

ial V

eloc

ity, vls

m/s

Gas Superfitial Velocity, vgs m/s

Slug

Dispersed Bubble

Dispersed Bubble

Elongated Buuble / PlugSlug

Anular

WavyStatified

0.001

0.01

0.1

1

10

0.01 0.10 1.00 10.00 100.00

Supe

rfiti

al L

iqui

d (w

ater

) Vel

ocity

, vls

m/s

Superfitial Gas Velocity, vgs m/s

SlugDispersed Bubble

Dispersed Bubble

Elongated Buuble / Plug Slug

Anular

WavyStatified

Formatted: Font: 10 pt

Formatted: Font: 10 pt

Figure 7. Comparison of the Mandhane et al. [9] (adapted) flow regime map with experimental dataobtained for horizontal gas/Newtonian flow. Copyright Elsevier, 1974.

3.1.2. Air/non-Newtonian flow map

Researchers have also developed different flow pattern maps for horizontal, vertical, and inclinedgas/non-Newtonian flow. For horizontal gas/non-Newtonian fluid, Chhabra & Richardson [56]developed a flow pattern map by slightly modifying the Mandhane et al. [9] horizontal flow patternmap (Figure 8). This map was developed for evaluating the literature and verified using 3700 datapoints of gas/non-Newtonian shear-thinning liquid mixture flow with 70% certainty. However, therewas not enough data to confirm the Chhabra & Richardson [56] flow map for annular and slug flows.

In Figure 8, the experimental flow regime almost matches with the Chhabra & Richardson [10]flow map, but the slug to dispersed bubble flow transition started slightly earlier for this experiment.Chhabra & Richardson [10] used the particulate suspension of china clay, aqueous polymer solutions,limestone, and coal, which is a significantly more viscous shear-thinning non-Newtonian fluidcompared to the 0.1% solution of xanthan gum that was used in this experiment. This is why thedispersed bubble flow regime started earlier. It is well-known that flow patterns of gas/non-Newtonianfluid are not significantly different from gas/Newtonian fluid for horizontal flow. However, due to highviscosity, the bubbles and slug could not break easily and collided together to form more prominentand well-defined bubbles.

Appl. Sci. 2020, 10, 3272 12 of 27

Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 28

there was not enough data to confirm the Chhabra & Richardson [56] flow map for annular and slug flows.

Figure 8: Comparison of the Chhabra & Richardson [10] (adapted) flow regime map with experimental data obtained for horizontal gas/non-Newtonian flow.

In Figure 8, the experimental flow regime almost matches with the Chhabra & Richardson [10] flow map, but the slug to dispersed bubble flow transition started slightly earlier for this experiment. Chhabra & Richardson [10] used the particulate suspension of china clay, aqueous polymer solutions, limestone, and coal, which is a significantly more viscous shear-thinning non-Newtonian fluid compared to the 0.1% solution of xanthan gum that was used in this experiment. This is why the dispersed bubble flow regime started earlier. It is well-known that flow patterns of gas/non-Newtonian fluid are not significantly different from gas/Newtonian fluid for horizontal flow. However, due to high viscosity, the bubbles and slug could not break easily and collided together to form more prominent and well-defined bubbles.

The experimental data were used to reconstruct the flow maps and validated with the existing literature for the identification of the two-phase flow regimes for this experimental setup. Due to the targeted velocity condition pre-requisite for hole cleaning operations and the limitation of the existing flow loop system (that is, not designed for the extreme conditions of higher flow rates and high-pressure environments), the experiment was performed under moderate pressure and velocity conditions. Therefore, the results are effectively limited by the bubble flow and plug flow range, which can provide stratified, wavy, or annular flow. The Taitel & Dukler [7] and Mandhane et al. [9] flow maps for air/water two-phase horizontal flow and the Chhabra & Richardson [56] flow map for air/xanthan gum solution horizontal two-phase flow represented the flow regimes of the experimental setup relatively accurately. Nonetheless, the boundary of the flow regime varied due to the unpredictable characteristics of the transition of the flow pattern. 3.1.3. Air/Newtonian Two-phase flow

The flow behavior of the air/Newtonian two-phase flow is shown in Figure 9. Figure 10 demonstrates that slug frequency increases with the increase of liquid superficial velocity for all studied combinations, while the superficial gas velocity was held constant for each set of data. Similar behavior was reported by Abed and Ghoben [22] for a Newtonian fluid (water). This happened due

0.001

0.01

0.1

1

10

0.01 0.10 1.00 10.00 100.00

Supe

rfitia

l Liq

uid

(non

-New

toni

an) V

eloc

ity,

v lns

m/s

Superfitial Gas Velocity, vgs m/s

SlugDispersed BubbleElongated bubble

Dispersed

Elongated Buuble / PlugSlug Anular

WavyStatified

Experimental Bounday Line

Figure 8. Comparison of the Chhabra & Richardson [10] (adapted) flow regime map with experimentaldata obtained for horizontal gas/non-Newtonian flow. Copyright Canadian Journal of ChemicalEngineering, 2020.

The experimental data were used to reconstruct the flow maps and validated with the existingliterature for the identification of the two-phase flow regimes for this experimental setup. Due tothe targeted velocity condition pre-requisite for hole cleaning operations and the limitation of theexisting flow loop system (that is, not designed for the extreme conditions of higher flow rates andhigh-pressure environments), the experiment was performed under moderate pressure and velocityconditions. Therefore, the results are effectively limited by the bubble flow and plug flow range, whichcan provide stratified, wavy, or annular flow. The Taitel & Dukler [7] and Mandhane et al. [9] flow mapsfor air/water two-phase horizontal flow and the Chhabra & Richardson [56] flow map for air/xanthangum solution horizontal two-phase flow represented the flow regimes of the experimental setuprelatively accurately. Nonetheless, the boundary of the flow regime varied due to the unpredictablecharacteristics of the transition of the flow pattern.

3.1.3. Air/Newtonian Two-phase Flow

The flow behavior of the air/Newtonian two-phase flow is shown in Figure 9. Figure 10 demonstratesthat slug frequency increases with the increase of liquid superficial velocity for all studied combinations,while the superficial gas velocity was held constant for each set of data. Similar behavior was reported byAbed and Ghoben [22] for a Newtonian fluid (water). This happened due to the increase in liquid volumefraction. The liquid occupies more space in the liquid film region as the elongated bubble unit becomessmaller, which is why the slug unit increased in number.

Figure 10 shows the effect of a superficial gas ratio on slug frequency. Results from Figure 10 revealthat, for a constant liquid flow rate, slug frequency decreases with increasing gas velocity. The slugfrequency initially decreases with the gas velocity up to around 5 m/s gas velocity, and then increasesagain, indicating additional slug formation. Moreover, to provide further insight, Figure 11 representsthe slug frequency vs. the mixture velocity for air/water flow.

Appl. Sci. 2020, 10, 3272 13 of 27

Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 28

to the increase in liquid volume fraction. The liquid occupies more space in the liquid film region as the elongated bubble unit becomes smaller, which is why the slug unit increased in number.

Figure 9: Effect of liquid superficial velocity on slug frequency for air/water flow.

Figure 10 shows the effect of a superficial gas ratio on slug frequency. Results from Figure 10 reveal that, for a constant liquid flow rate, slug frequency decreases with increasing gas velocity. The slug frequency initially decreases with the gas velocity up to around 5 m/s gas velocity, and then increases again, indicating additional slug formation. Moreover, to provide further insight, Figure 11 represents the slug frequency vs. the mixture velocity for air/water flow.

Figure 10: Effect of gas superficial velocity on slug frequency of air/water two-phase flow.

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

1.5 1.7 1.9 2.1 2.3 2.5

Slug

Fre

quen

cy f s

, 1/s

Liquid velocity, ls, m/s

vg=2.8 m/svg=3.7 m/svg= 4.7 m/svg= 6.4 m/s

0

0.5

1

1.5

2

2.5

3

1.5 2.5 3.5 4.5 5.5 6.5 7.5

Slug

Fre

quen

cy f s

, 1/s

Gas velocity, ls, m/s

vl=1.5m/svl=1.76m/svl=1.95m/svl=2.3m/s

Figure 9. Effect of liquid superficial velocity on slug frequency for air/water flow.Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 31

Figure 10. Effect of gas superficial velocity on slug frequency of air/water two-phase flow.

Figure 11. Slug frequency vs. mixture velocity for air/water flow.

Comparing Figures 9 and 11, it is evident that the slug frequency curves mainly depend on superficial gas velocity. Two of these graphs also show that in the range of 5 m/s to 6.5 m/s, the slug frequency reaches a minimum, and the slug frequency increases with increasing mixture velocity or gas superficial velocity. The sudden increase in slug frequency perhaps indicates the transition of the flow regime from slug to dispersed bubble flow. The higher gas flow rates induce more turbulence in the fluid flow, initiating the slug breakdown, which is attributed to the increase in the overall number of slugs in the system. It has also been observed that the number of dispersed bubbles

Figure 10. Effect of gas superficial velocity on slug frequency of air/water two-phase flow.

Comparing Figures 9 and 11, it is evident that the slug frequency curves mainly depend onsuperficial gas velocity. Two of these graphs also show that in the range of 5 m/s to 6.5 m/s, the slugfrequency reaches a minimum, and the slug frequency increases with increasing mixture velocity orgas superficial velocity. The sudden increase in slug frequency perhaps indicates the transition of theflow regime from slug to dispersed bubble flow. The higher gas flow rates induce more turbulence inthe fluid flow, initiating the slug breakdown, which is attributed to the increase in the overall numberof slugs in the system. It has also been observed that the number of dispersed bubbles increases in

Appl. Sci. 2020, 10, 3272 14 of 27

the slug pocket and wet film area. This indicates the beginning of the transition of the flow pattern.These findings also agree with the experimental data of Otten & Fayed [40] and Gregory & Scott [41].Moreover, the slope of slug frequency vs. slug Froude number is reported in Figure 12.Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 28

Figure 11: Slug frequency vs. mixture velocity for air/water flow.

Comparing Figure 9 and Figure 11, it is evident that the slug frequency curves mainly depend on superficial gas velocity. Two of these graphs also show that in the range of 5 m/s to 6.5 m/s, the slug frequency reaches a minimum, and the slug frequency increases with increasing mixture velocity or gas superficial velocity. The sudden increase in slug frequency perhaps indicates the transition of the flow regime from slug to dispersed bubble flow. The higher gas flow rates induce more turbulence in the fluid flow, initiating the slug break down, which is attributed to the increase in the overall number of slugs in the system. It has also been observed that the number of dispersed bubbles increases in the slug pocket and wet film area. This indicates the beginning of the transition of the flow pattern. These findings also agree with the experimental data of Otten & Fayed [40] and Gregory & Scott [41]. Moreover, the slope of slug frequency vs. slug Froude number is reported in Figure 12.

Figure 12: Slug frequency vs. Froude number for air/water flow.

y = 0.0673x0.9757

1

1.4

1.8

2.2

2.6

3

20 25 30 35 40 45

Slug

Fre

quen

cy f s

, 1/s

Froude Number, Nfs

0

0.5

1

1.5

2

2.5

3

1.5 3.5 5.5 7.5 9.5

Slug

Fre

quen

cy f s

, 1/s

Mixture velocity, m, m/s

vl=1.5 m/s

vl=1.7 m/s

Figure 11. Slug frequency vs. mixture velocity for air/water flow.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 31

increases in the slug pocket and wet film area. This indicates the beginning of the transition of the flow pattern. These findings also agree with the experimental data of Otten & Fayed [40] and Gregory & Scott [41]. Moreover, the slope of slug frequency vs. slug Froude number is reported in Figure 12.

Figure 12. Slug frequency vs. Froude number for air/water flow.

In Figure 12, the slope of slug frequency vs. slug Froude number yields an equation of =0.0673 .. This equation shows a slight deviation from Gregory & Scott [41], which is shown

in Equation (5), perhaps due to the experimental conditions and the assumption (vm = vs) for the air–water flow of this experiment. To evaluate the standard deviation and experimental accuracy of this study, Figure 13 reports the regression analysis of slug frequency and Froude number.

Figure 12. Slug frequency vs. Froude number for air/water flow.

In Figure 12, the slope of slug frequency vs. slug Froude number yields an equation of fs =

0.0673(

N fr

)0.098. This equation shows a slight deviation from Gregory & Scott [41], which is shown

in Equation (5), perhaps due to the experimental conditions and the assumption (vm = vs) for theair–water flow of this experiment. To evaluate the standard deviation and experimental accuracy ofthis study, Figure 13 reports the regression analysis of slug frequency and Froude number.

Appl. Sci. 2020, 10, 3272 15 of 27

Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 29

In Figure 12, the slope of slug frequency vs. slug Froude number yields an equation of =0.0673 .. This equation shows a slight deviation from Gregory & Scott [41], which is shown

in Equation (5), perhaps due to the experimental conditions and the assumption (vm = vs) for the air–water flow of this experiment. To evaluate the standard deviation and experimental accuracy of this study, Figure 13 reports the regression analysis of slug frequency and Froude number.

Figure 13: Relationship between slug frequency and Froude number for air–water system data (R2 = 88.1%). From Figure 13, the R2 value was found to be 88.1%, which indicates that the slug frequency and

Froude number have a good relationship. Since the experimental and modeled data are close to the

regression line, the equation = 0.0673 . can explain the variability of the data around its

mean. In order to further validate and test the reliability of the obtained experimental data, the results

are reported with correlations with different model results reported in various previous studies. Figure 14 shows the experimental and modeled validation with the Gregory & Scott [41] correlation model for air–water two-phase flow.

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

20 25 30 35 40 45

Slug

Fre

quen

cy f s

, 1/s

Froude Number, Nfs

Model(Frequency)Conf. interval (Obs 95%)

Figure 13. Relationship between slug frequency and Froude number for air–water system data (R2 = 88.1%).

From Figure 13, the R2 value was found to be 88.1%, which indicates that the slug frequencyand Froude number have a good relationship. Since the experimental and modeled data are close to

the regression line, the equation fs = 0.0673(

N fr

)0.098can explain the variability of the data around

its mean.In order to further validate and test the reliability of the obtained experimental data, the results are

reported with correlations with different model results reported in various previous studies. Figure 14shows the experimental and modeled validation with the Gregory & Scott [41] correlation model forair–water two-phase flow.Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 29

Figure 14: Experimental slug frequency for the air–water system compared with the prediction model of Gregory & Scott [41] correlation (R2=73.8%).

From Figure 14, it is observed that all the data points are close to the regression line, the regression has an R2 value of 73.8%, and all the experimental data fit well within in the 95% confidence interval. The experimental slug frequency results are also compared with the Zabaras et al. [23] predicted model for the air–water system in Figure 15. The results reveal that the data points of the Zabaras et al. [23] predicted model have comparatively deviated from the regression line, with an R2 value of 60.3% at a 95% confidence interval.

Figure 15: Experimental slug frequency for the air–water system compared with the prediction model of Zabaras et al. [23] correlation (R2=60%).

The experimental data and the predictions of slug frequency of the Gregory & Scott [41] model have an R2 value of 73.8%, and the Zabaras et al. [23] model has an R2 value of 60%. Therefore, the

0

0.5

1

1.5

2

2.5

1 1.5 2 2.5 3

Pred

icte

d f s

, 1/s

Experimental fs , 1/s

Model(Predicted)Conf. interval (Obs 95%)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1 1.5 2 2.5 3

Pred

icte

d f s

, 1/s

Experimental fs , 1/s

Model(Predicted)Conf. interval (Obs 95%)

Figure 14. Experimental slug frequency for the air–water system compared with the prediction modelof Gregory & Scott [41] correlation (R2 = 73.8%).

Appl. Sci. 2020, 10, 3272 16 of 27

From Figure 14, it is observed that all the data points are close to the regression line, the regressionhas an R2 value of 73.8%, and all the experimental data fit well within in the 95% confidence interval.The experimental slug frequency results are also compared with the Zabaras et al. [23] predicted modelfor the air–water system in Figure 15. The results reveal that the data points of the Zabaras et al. [23]predicted model have comparatively deviated from the regression line, with an R2 value of 60.3% at a95% confidence interval.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 28

Figure 14: Experimental slug frequency for the air–water system compared with the prediction model of Gregory & Scott [41] correlation (R2=73.8%).

From Figure 14, it is observed that all the data points are close to the regression line, the regression has an R2 value of 73.8%, and all the experimental data fit well within in the 95% confidence interval. The experimental slug frequency results are also compared with the Zabaras et al. [23] predicted model for the air–water system in Figure 15. The results reveal that the data points of the Zabaras et al. [23] predicted model have comparatively deviated from the regression line, with an R2 value of 60.3% at a 95% confidence interval.

Figure 15: Experimental slug frequency for the air–water system compared with the prediction model of Zabaras et al. [23] correlation (R2=60%).

The experimental data and the predictions of slug frequency of the Gregory & Scott [41] model have an R2 value of 73.8%, and the Zabaras et al. [23] model has an R2 value of 60%. Therefore, the Gregory & Scott [41] model is found to be closer to the experimental data. In the above graphs, the differences between experimental and predicted data varied mainly due to the variation in experimental conditions and setups [22]. Since the experimental data are more similar to the Gregory & Scott [41] model, Figure 16 shows the Gregory & Scott [41] correlation with the standard deviation of three samples under the same experimental condition. The error bar indicates a 95% confidence interval of the data, and none of the confidence intervals includes zero, which means the data are statistically significant and repeatable for air/water two-phase flow.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1 1.5 2 2.5 3

Pred

icte

d f s

, 1/s

Experimental fs , 1/s

Model(Predicted)Conf. interval (Obs 95%)

Figure 15. Experimental slug frequency for the air–water system compared with the prediction modelof Zabaras et al. [23] correlation (R2 = 60%).

The experimental data and predictions of slug frequency of the Gregory & Scott [41] model havean R2 value of 73.8%, and the Zabaras et al. [23] model has an R2 value of 60%. Therefore, the Gregory& Scott [41] model is found to be closer to the experimental data. In the above graphs, the differencesbetween experimental and predicted data varied mainly due to the variation in experimental conditionsand setups [22]. Since the experimental data are more similar to the Gregory & Scott [41] model,Figure 16 shows the Gregory & Scott [41] correlation with the standard deviation of three samplesunder the same experimental condition. The error bar indicates a 95% confidence interval of the data,and none of the confidence intervals includes zero, which means the data are statistically significantand repeatable for air/water two-phase flow.Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 28

Figure 16: Experimental slug frequency for the air–water system compared to the predictions of Gregory & Scott [41] correlation with standard deviations.

3.1.4. Air/non-Newtonian Two-phase flow In this air/non-Newtonian fluid experiment, a 0.1% solution of xanthan gum was used as the

non-Newtonian fluid. Non-Newtonian fluids are generally superior drilling fluids compared to Newtonian fluids due to their viscosity and the holding/uplifting capacity of the cuttings. Figure 17 illustrates the impact of liquid superficial velocity on slug frequency for the air/non-Newtonian flow system.

Figure 17: Effect of liquid superficial velocity on slug frequency for air/non-Newtonian flow.

In Figure 17, the slug frequency increases with the increment of liquid non-Newtonian superficial velocity when the superficial gas velocity is held constant. Therefore, at lower superficial liquid velocity, the slug frequency increases sharply, and at higher liquid velocity, the slug frequency

R² = 0.7214

0

0.5

1

1.5

2

2.5

0 1 2 3 4

Pred

icte

d f s

, 1/s

Experimental fs , 1/s

11.21.41.61.8

22.22.42.62.8

3

1.5 1.7 1.9 2.1 2.3 2.5

Slug

Fre

quen

cy f s

n, 1/

s

Xanthan gum velocity, ln, m/s

vg=2.9 m/svg=3.7 m/svg=4.5 m/svg=6.5 m/s

Commented [MW1]: Please confirm intended meaning is

retained.

Figure 16. Experimental slug frequency for the air–water system compared to the predictions ofGregory & Scott [41] correlation with standard deviations.

Appl. Sci. 2020, 10, 3272 17 of 27

3.1.4. Air/non-Newtonian Two-Phase Flow

In this air/non-Newtonian fluid experiment, a 0.1% solution of xanthan gum was used as thenon-Newtonian fluid. Non-Newtonian fluids are generally superior drilling fluids compared toNewtonian fluids due to their viscosity and the holding/uplifting capacity of the cuttings. Figure 17illustrates the impact of liquid superficial velocity on slug frequency for the air/non-Newtonianflow system.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 28

Figure 16: Experimental slug frequency for the air–water system compared to the predictions of Gregory & Scott [41] correlation with standard deviations.

3.1.4. Air/non-Newtonian Two-phase flow In this air/non-Newtonian fluid experiment, a 0.1% solution of xanthan gum was used as the

non-Newtonian fluid. Non-Newtonian fluids are generally superior drilling fluids compared to Newtonian fluids due to their viscosity and the holding/uplifting capacity of the cuttings. Figure 17 illustrates the impact of liquid superficial velocity on slug frequency for the air/non-Newtonian flow system.

Figure 17: Effect of liquid superficial velocity on slug frequency for air/non-Newtonian flow.

In Figure 17, the slug frequency increases with the increment of liquid non-Newtonian superficial velocity when the superficial gas velocity is held constant. Therefore, at lower superficial liquid velocity, the slug frequency increases sharply, and at higher liquid velocity, the slug frequency

R² = 0.7214

0

0.5

1

1.5

2

2.5

0 1 2 3 4

Pred

icte

d f s

, 1/s

Experimental fs , 1/s

11.21.41.61.8

22.22.42.62.8

3

1.5 1.7 1.9 2.1 2.3 2.5

Slug

Fre

quen

cy f s

n, 1/

s

Xanthan gum velocity, ln, m/s

vg=2.9 m/svg=3.7 m/svg=4.5 m/svg=6.5 m/s

Commented [MW1]: Please confirm intended meaning is

retained.

Figure 17. Effect of liquid superficial velocity on slug frequency for air/non-Newtonian flow.

In Figure 17, the slug frequency increases with the increment of liquid non-Newtonian superficialvelocity when the superficial gas velocity is held constant. Therefore, at lower superficial liquid velocity,the slug frequency increases sharply, and at higher liquid velocity, the slug frequency decreases. As theliquid velocity increases, the air requires more energy to drive the viscous fluid; nevertheless, the airflowrate is constant for each set. This explains why the number of slugs decreases as the liquid velocityincreases at continuous airflow (see Figure 18). Furthermore, Figure 18 highlights the relationshipbetween gas superficial velocity and slug frequency for air/non-Newtonian flow conditions.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 18 of 28

decreases. As the liquid velocity increases, the air requires more energy to drive the viscous fluid; nevertheless, the airflow rate is constant for each set. This explains why the number of slugs decreases as the liquid velocity increases at continuous airflow (see Figure 18). Furthermore, Figure 18 highlights the relationship between gas superficial velocity and slug frequency for air/non-Newtonian flow conditions.

Figure 18: Effect of gas superficial velocity on slug frequency for air/non-Newtonian flow.

In Figure 18, slug frequency change is shown with superficial gas velocity for a constant liquid superficial velocity. Here, 0.1% xanthan gum solution was used as a non-Newtonian fluid where the power-law index, n, is equal to 0.81 and k = 0.009344. From Figure 18 it is apparent that the slug frequency decreases as the gas velocity rises to 6.5 m/s. Because the gas flow rate increases in a constant liquid velocity, Taylor bubble formation becomes relatively more significant; therefore, the length of the slug unit increases, and slug frequency decreases. The evaluation of the slug frequency vs. mixture velocity for air/non-Newtonian fluid flow is reported in Figure 19.

Figure 19: Slug frequency vs. mixture velocity for air/non-Newtonian fluid flow.

0

0.5

1

1.5

2

2.5

3

3.5

1.5 3.5 5.5 7.5 9.5

Slug

Fre

quen

cy f s

n, 1/

s

Mixture velocity, vmn, m/s

vl=1.57 m/svl=1.76 m/svl=1.96 m/svl=2.15 m/svl=2.36 m/s

0

0.5

1

1.5

2

2.5

3

3.5

1 2 3 4 5 6 7

Slug

Fre

quen

cy f s

n, 1/

s

Gas velocity gn, m/s

vl=1.57 m/svl=1.76 m/svl=1.96 m/svl=2.15 m/svl=2.5 m/s

Figure 18. Effect of gas superficial velocity on slug frequency for air/non-Newtonian flow.

In Figure 18, slug frequency change is shown with superficial gas velocity for a constant liquidsuperficial velocity. Here, 0.1% xanthan gum solution was used as a non-Newtonian fluid where thepower-law index, n, is equal to 0.81 and k = 0.009344. From Figure 18 it is apparent that the slugfrequency decreases as the gas velocity rises to 6.5 m/s. Because the gas flow rate increases in a constant

Appl. Sci. 2020, 10, 3272 18 of 27

liquid velocity, Taylor bubble formation becomes relatively more significant; therefore, the length ofthe slug unit increases, and slug frequency decreases. The evaluation of the slug frequency vs. mixturevelocity for air/non-Newtonian fluid flow is reported in Figure 19.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 18 of 28

decreases. As the liquid velocity increases, the air requires more energy to drive the viscous fluid; nevertheless, the airflow rate is constant for each set. This explains why the number of slugs decreases as the liquid velocity increases at continuous airflow (see Figure 18). Furthermore, Figure 18 highlights the relationship between gas superficial velocity and slug frequency for air/non-Newtonian flow conditions.

Figure 18: Effect of gas superficial velocity on slug frequency for air/non-Newtonian flow.

In Figure 18, slug frequency change is shown with superficial gas velocity for a constant liquid superficial velocity. Here, 0.1% xanthan gum solution was used as a non-Newtonian fluid where the power-law index, n, is equal to 0.81 and k = 0.009344. From Figure 18 it is apparent that the slug frequency decreases as the gas velocity rises to 6.5 m/s. Because the gas flow rate increases in a constant liquid velocity, Taylor bubble formation becomes relatively more significant; therefore, the length of the slug unit increases, and slug frequency decreases. The evaluation of the slug frequency vs. mixture velocity for air/non-Newtonian fluid flow is reported in Figure 19.

Figure 19: Slug frequency vs. mixture velocity for air/non-Newtonian fluid flow.

0

0.5

1

1.5

2

2.5

3

3.5

1.5 3.5 5.5 7.5 9.5

Slug

Fre

quen

cy f s

n, 1/

s

Mixture velocity, vmn, m/s

vl=1.57 m/svl=1.76 m/svl=1.96 m/svl=2.15 m/svl=2.36 m/s

0

0.5

1

1.5

2

2.5

3

3.5

1 2 3 4 5 6 7

Slug

Fre

quen

cy f s

n, 1/

s

Gas velocity gn, m/s

vl=1.57 m/svl=1.76 m/svl=1.96 m/svl=2.15 m/svl=2.5 m/s

Figure 19. Slug frequency vs. mixture velocity for air/non-Newtonian fluid flow.

Figure 19 represents the change of slug frequency with the mixture velocity of air–xanthangum flow. In addition, it is seen that until 6.5 m/s, the mixture velocity slug frequency is minimal.Otten & Fayed [40] also obtained similar patterns for air/non-Newtonian flow. The same phenomenaalso occurred in Figure 11 for gas/Newtonian flow. However, the minimum slug frequency was around5 m/s mixture velocity, which happened significantly earlier than the gas/non-Newtonian two-phaseflow. Here, we can observe the inevitable effect of viscosity. Water viscosity at 20 ◦C room temperatureis around 1 cP, and the experimental viscosity of 0.1% xanthan gum is 2.4 cP, which is slightly moreviscous than water.

The flow mechanism of slug flow is that the gas bubble is trapped in the water and drives thewater forward almost at the same velocity as the gas velocity. Nevertheless, when the liquid becomesviscous, the gas requires more energy to drive the liquid forward. At a constant airflow rate, it is hardto achieve extra energy, so the whole process becomes slow, and the slug velocity and number of slugsdecrease [42]. If further experiments were conducted for gas/non-Newtonian fluids, slug frequencywould increase again, with an increased gas flow rate in the slug to bubbly flow transition zone as thegas/water two-phase flow results in the transition from slug to dispersed bubble flow. The turbulenceof the flow structure begins to increase, the unit slug starts to break down, and the number of slugsincreases at higher gas flow rates. It has also been observed that the number of dispersed bubblesincreases in the slug pocket and wet film area. This indicates the beginning of the transition of theflow pattern.

To understand this result further and demonstrate its wider applicability, the present studywas extended to a non-dimensional number, i.e., the Froude number, presented in Figure 20 as therelationship between slug frequencies and Froude number for an air/xanthan gum solution.

In Figure 20, it is shown that the slope of slug frequency vs. slug Froude number for air/xanthan

gum solution can be modeled using the equation vs = 0.0082(

N fr

)1.5624, where the model strength R2

is 80.62%. In order to further validate the experimental data for an air/non-Newtonian two-phase flowsystem, Figures 21 and 22 represent the relationship between the experimental and predicted data of theGregory & Scott [41] proposed model with standard error bars and 95% confidence level, respectively.

Appl. Sci. 2020, 10, 3272 19 of 27

Appl. Sci. 2020, 10, x FOR PEER REVIEW 19 of 28

Figure 19 represents the change of slug frequency with the mixture velocity of air–xanthan gum flow. In addition, it is seen that until 6.5 m/s, the mixture velocity slug frequency is minimal. Otten & Fayed [40] also obtained similar patterns for air/non-Newtonian flow. The same phenomena also occurred in Figure 11 for gas/Newtonian flow. However, the minimum slug frequency was around 5 m/s mixture velocity, which happened significantly earlier than the gas/non-Newtonian two-phase flow. Here, we can observe the inevitable effect of viscosity. Water viscosity at 20 °C room temperature is around 1 cP, and the experimental viscosity of 0.1% xanthan gum is 2.4 cP, which is slightly more viscous than water.

The flow mechanism of slug flow is that the gas bubble is trapped between water and drives water forward almost at the same velocity as the gas velocity. Nevertheless, when the liquid becomes viscous, the gas requires more energy to drive the liquid forward. At a constant airflow rate, it is hard to achieve extra energy, so the whole process becomes slow, and the slug velocity and number of slugs decrease [42]. If further experiments were conducted for gas/non-Newtonian fluids, slug frequency would increase again, with an increased gas flow rate in the slug to bubbly flow transition zone as the gas/water two-phase flow results in the transition from slug to dispersed bubble flow. The turbulence of the flow structure begins to increase, the unit slug starts to break down, and the number of slugs increases at higher gas flow rates. It has also been observed that the number of dispersed bubbles increases in the slug pocket and wet film area. This indicates the beginning of the transition of the flow pattern.

To understand this result further and demonstrate its wider applicability, the present study was extended to a non-dimensional number, i.e., the Froude number, presented in Figure 20 as the relationship between slug frequencies and Froude number for an air/xanthan gum solution.

Figure 20: Slug frequency vs. Froude number for an air/xanthan gum solution.

In Figure 20, it is shown that the slope of slug frequency vs. slug Froude number for air/xanthan

gum solution can be modeled using the equation = 0.0082 ., where the model strength

R2 is 80.62%. In order to further validate the experimental data for an air/non-Newtonian two-phase flow system, Figure 21 and Figure 22 represent the relationship between the experimental and predicted data of the Gregory & Scott [41] proposed model with standard error bars and 95% confidence level, respectively.

y = 0.0082x1.5624

R² = 0.8062

0

0.5

1

1.5

2

2.5

3

3.5

20 25 30 35 40 45

Slug

Fre

quen

cy f s

n, 1/

s

Froude Number, Nfrn

Figure 20. Slug frequency vs. Froude number for an air/xanthan gum solution.Appl. Sci. 2020, 10, x FOR PEER REVIEW 20 of 28

Figure 21: Experimental vs. predicted data [41] with standard error bars.

The error bar in Figure 21 shows the 95% confidence interval of the data; none of the confidence intervals includes zero, which means the data are statistically significant and repeatable for air/water two-phase flow.

Figure 22: Experimental slug frequency for air–xanthan gum system compared to the predictions by the Gregory & Scott [41] correlation, where R2 = 74.6%.

Picchi et al. [47] modified the slug frequency equation of Gregory & Scott [41] for a shear-thinning non-Newtonian fluid. Figure 21 represents the comparison of experimental results with the modified Gregory & Scott [41] slug frequency equation with error bars. The R2 value of 75.3% also

R² = 0.753

0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

2.3

0.5 1 1.5 2 2.5 3

Pred

icte

d f s

, 1/s

Experimental fs , 1/s

0.9

1.1

1.3

1.5

1.7

1.9

2.1

2.3

1 1.5 2 2.5 3 3.5

Slug

Fre

quen

cy f s

n, 1/

s

Experimental fsn , 1/s

Model(fs)Conf. interval (Obs 95%)

Figure 21. Experimental vs. predicted data [41] with standard error bars.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 20 of 28

Figure 21: Experimental vs. predicted data [41] with standard error bars.

The error bar in Figure 21 shows the 95% confidence interval of the data; none of the confidence intervals includes zero, which means the data are statistically significant and repeatable for air/water two-phase flow.

Figure 22: Experimental slug frequency for air–xanthan gum system compared to the predictions by the Gregory & Scott [41] correlation, where R2 = 74.6%.

Picchi et al. [47] modified the slug frequency equation of Gregory & Scott [41] for a shear-thinning non-Newtonian fluid. Figure 21 represents the comparison of experimental results with the modified Gregory & Scott [41] slug frequency equation with error bars. The R2 value of 75.3% also

R² = 0.753

0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

2.3

0.5 1 1.5 2 2.5 3

Pred

icte

d f s

, 1/s

Experimental fs , 1/s

0.9

1.1

1.3

1.5

1.7

1.9

2.1

2.3

1 1.5 2 2.5 3 3.5

Slug

Fre

quen

cy f s

n, 1/

s

Experimental fsn , 1/s

Model(fs)Conf. interval (Obs 95%)

Figure 22. Experimental slug frequency for the air–xanthan gum system compared to predictions bythe Gregory & Scott [41] correlation, where R2 = 74.6%.

Appl. Sci. 2020, 10, 3272 20 of 27

The error bar in Figure 21 shows the 95% confidence interval of the data; none of the confidenceintervals includes zero, which means the data are statistically significant and repeatable for air/watertwo-phase flow.

Picchi et al. [47] modified the slug frequency equation of Gregory & Scott [41] for a shear-thinningnon-Newtonian fluid. Figure 21 represents the comparison of experimental results with the modifiedGregory & Scott [41] slug frequency equation with error bars. The R2 value of 75.3% also representsthe reliability and repeatability of the experimental data of this study. Figure 22 also shows the 95%confidence interval of the experimental data.

3.2. Wavelet Spectrum Analysis

The wavelet packet analysis decomposed the pressure signals into four levels. Among the waveletdecomposition methods, Daubechies’ four-scale base wavelet (db4) has been used most frequently inmultiphase flow time series decompaction [57]. In this study, Daubechies’ four-scale base wavelet (db4)and norm entropy analysis method generated 16 wavelet packet coefficients. The pressure fluctuationsignal achieved from the experimental data only yields a frequency of 100Hz. Thus, only four-leveldecomposition is sufficient because the pressure signals do not have sufficiently high-frequency andhigh-resolution data to obtain more detailed frequency analysis. The spectrum of the packet waveletanalysis represents the time-frequency plot, which provides a coefficient of decomposed frequencies ata different level. This spectrum represents the time and location of the fluctuation of the signal.

From Figure 23, it can be observed that the time-frequency plot divides the time-frequency planeinto concentrated rectangles as a two-dimensional representation of signals. The pink color intensity ofeach box depends on the coefficient of the wavelet packet [34].

Appl. Sci. 2020, 10, x FOR PEER REVIEW 22 of 31

modified Gregory & Scott [41] slug frequency equation with error bars. The R2 value of 75.3% also represents the reliability and repeatability of the experimental data of this study. Figure 22 also shows the 95% confidence interval of the experimental data.

3.2. Wavelet Spectrum Analysis

The wavelet packet analysis decomposed the pressure signals into four levels. Among the wavelet decomposition methods, Daubechies’ four-scale base wavelet (db4) has been used most frequently in multiphase flow time series decompaction [57]. In this study, Daubechies’ four-scale base wavelet (db4) and norm entropy analysis method generated 16 wavelet packet coefficients. The pressure fluctuation signal achieved from the experimental data only yields a frequency of 100Hz. Thus, only four-level decomposition is sufficient because the pressure signals do not have sufficiently high-frequency and high-resolution data to obtain more detailed frequency analysis. The spectrum of the packet wavelet analysis represents the time-frequency plot, which provides a coefficient of decomposed frequencies at a different level. This spectrum represents the time and location of the fluctuation of the signal.

Figure 23. Spectrum for slug flow under different flow conditions.

From Figure 23, it can be observed that the time-frequency plot divides the time-frequency plane into concentrated rectangles as a two-dimensional representation of signals. The pink color intensity of each box depends on the coefficient of the wavelet packet [34].

A similar plot is shown in Figure 24, which is the time-frequency plot for the bubbly flow regime for various flow conditions. The intensity of the pink shade represents the energy level of a time-

Figure 23. Spectrum for slug flow under different flow conditions.

Appl. Sci. 2020, 10, 3272 21 of 27

A similar plot is shown in Figure 24, which is the time-frequency plot for the bubbly flow regime forvarious flow conditions. The intensity of the pink shade represents the energy level of a time-frequencycell; the lower the energy content, the lighter the shade. For the bubbly flow regime, the bubbles aresmaller, so the pressure fluctuation intensity is less, which means the low-frequency response has lessenergy content and the pink shade is light and scattered. In the slug flow, the Taylor bubble size is moreprominent. Therefore, the low-frequency cells have more energy and a darker shade [34]. In addition,the repetition of the intense pink shade after a specific time interval can be evidence of the picks ofthe pressure signal. Thus, with a higher-resolution and better-quality sensor, where the pressuresignal picks are more precise, this map can be a helpful way to understand the flow phenomena insidethe pipe. When comparing the wavelet spectrum analysis of bubbly and slug flows for the samewater flow rate, it has been observed that, for bubbly flow, the color intensity is comparatively less inthe low-frequency response area. However, the use of a higher-resolution pressure transducer mayenhance the wavelet spectrum quality with more precise fluctuation characteristics.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 23 of 31

frequency cell; the lower the energy content, the lighter the shade. For the bubbly flow regime, the bubbles are smaller, so the pressure fluctuation intensity is less, which means the low-frequency response has less energy content and the pink shade is light and scattered. In the slug flow, the Taylor bubble size is more prominent. Therefore, the low-frequency cells have more energy and a darker shade [34]. In addition, the repetition of the intense pink shade after a specific time interval can be evidence of the picks of the pressure signal. Thus, with a higher-resolution and better-quality sensor, where the pressure signal picks are more precise, this map can be a helpful way to understand the flow phenomena inside the pipe. When comparing the wavelet spectrum analysis of bubbly and slug flows for the same water flow rate, it has been observed that, for bubbly flow, the color intensity is comparatively less in the low-frequency response area. However, the use of a higher-resolution pressure transducer may enhance the wavelet spectrum quality with more precise fluctuation characteristics.

Figure 24. Spectrum for bubbly flow under different flow conditions.

Wavelet Entropy Analysis

The wavelet entropy analysis of the pressure fluctuation data represents the nonlinearity of the gas/liquid two-phase flow. After calculating the wavelet entropy of 10,000 pressure signal data points of gas/liquid two-phase horizontal flow, it was seen that the wavelet entropy increased with the increase of the pressure signal fluctuation.

The entropy values of the pressure fluctuation data were compared with the gas to liquid volume flowrate ratio (GLR) and the void fraction ( = ). The void fraction ( g) is the ratio of

the gas velocity and the liquid velocity. In Figure 25, it can also be observed that wavelet entropy increased with the increase of gas void

fraction for gas/non-Newtonian fluid. This means the fluctuation of the pressure increases with the

Figure 24. Spectrum for bubbly flow under different flow conditions.

Wavelet Entropy Analysis

The wavelet entropy analysis of the pressure fluctuation data represents the nonlinearity of thegas/liquid two-phase flow. After calculating the wavelet entropy of 10,000 pressure signal data points ofgas/liquid two-phase horizontal flow, it was seen that the wavelet entropy increased with the increaseof the pressure signal fluctuation.

The entropy values of the pressure fluctuation data were compared with the gas to liquid volume

flowrate ratio (GLR) and the void fraction (αg =vs

gvs

g+vsl). The void fraction (αg) is the ratio of the gas

velocity and the liquid velocity.

Appl. Sci. 2020, 10, 3272 22 of 27

In Figure 25, it can also be observed that wavelet entropy increased with the increase of gasvoid fraction for gas/non-Newtonian fluid. This means the fluctuation of the pressure increases withthe rise in the GLR for gas/water flow. Another observation is that the wavelet entropy value forgas/non-Newtonian flow is less than that for the gas/Newtonian fluid flow. This phenomenon occurreddue to the viscosity effect of the non-Newtonian fluid, which means the disturbance of the pressuresignals is less than that of the Newtonian fluid.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 22 of 28

where the pressure signal picks are more precise, this map can be a helpful way to understand the flow phenomena inside the pipe. When comparing the wavelet spectrum analysis of bubbly and slug flows for the same water flow rate, it has been observed that, for bubbly flow, the color intensity is comparatively less in the low-frequency response area. However, the use of a higher-resolution pressure transducer may enhance the wavelet spectrum quality with more precise fluctuation characteristics. 3.2.1. Wavelet Entropy Analysis

The wavelet entropy analysis of the pressure fluctuation data represents the nonlinearity of the gas/liquid two-phase flow. After calculating the wavelet entropy of 10,000 pressure signal data points of gas/liquid two-phase horizontal flow, it was seen that the wavelet entropy increased with the increase of the pressure signal fluctuation.

The entropy values of the pressure fluctuation data were compared with the gas to liquid volume flowrate ratio (GLR) and the void fraction ( = ). The void fraction ( g) is the ratio

of the gas velocity and the liquid velocity.

Figure 25: Change of wavelet entropy with gas void fraction for gas/Newtonian fluid.

In Figure 25, it can also be observed that wavelet entropy increased with the increase of gas void fraction for gas/non-Newtonian fluid. This means the fluctuation of the pressure increases with the rise in the GLR for gas/water flow. Another observation is that the wavelet entropy value for gas/non-Newtonian flow is less than that for the gas/Newtonian fluid flow. This phenomenon occurred due to the viscosity effect of the non-Newtonian fluid, which means the disturbance of the pressure signals is less than that of the Newtonian fluid.

Figure 26 shows that with the growth of the GLR, the norm wavelet entropy increased, which means the fluctuation of the pressure increases with the increase of the GLR. However, the norm entropy change at the low GLR is not consistent. Fan et al. (2013) also saw similar behavior for low GLR values, mostly in the bubble flow or bubble-slug transition flow region. As bubble flow and motion are random and complicated, it is hard for low-resolution sensors or wavelet norm entropy to detect the change. Figure 27 and Figure 28 show the wavelet norm entropy of bubble, bubble-elongated bubble, and slug flow regimes at different GLR combinations for the experimental setup used in this study.

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

Wav

elet

Ent

ropy

Void Fraction, g

vl=1.56 m/svl=1.76 m/svl=1.95 m/s

Figure 25. Change of wavelet entropy with gas void fraction for gas/Newtonian fluid.

Figure 26 shows that with the growth of the GLR, the norm wavelet entropy increased, whichmeans the fluctuation of the pressure increases with the increase of the GLR. However, the normentropy change at the low GLR is not consistent. Fan et al. (2013) also saw similar behavior for low GLRvalues, mostly in the bubble flow or bubble-slug transition flow region. As bubble flow and motionare random and complicated, it is hard for low-resolution sensors or wavelet norm entropy to detectthe change. Figures 27 and 28 show the wavelet norm entropy of bubble, bubble-elongated bubble,and slug flow regimes at different GLR combinations for the experimental setup used in this study.Appl. Sci. 2020, 10, x FOR PEER REVIEW 23 of 28

Figure 27: Wavelet entropy flow map for gas/Newtonian flow.

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5

Wav

elet

Ent

ropy

GLR

BubbleSlugBubble-Elongated bubble

Figure 26: Change of wavelet entropy with gas to liquid ratio for gas/Newtonian flow.

0

1

2

3

4

5

6

0 1 2 3 4 5

Wav

elet

Ent

ropy

GLR

vl=1.56 m/svl=1.76 m/svl=1.95 m/s

Figure 26. Change of wavelet entropy with gas to liquid ratio for gas/Newtonian flow.

Appl. Sci. 2020, 10, 3272 23 of 27

Appl. Sci. 2020, 10, x FOR PEER REVIEW 23 of 28

Figure 27: Wavelet entropy flow map for gas/Newtonian flow.

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5

Wav

elet

Ent

ropy

GLR

BubbleSlugBubble-Elongated bubble

Figure 26: Change of wavelet entropy with gas to liquid ratio for gas/Newtonian flow.

0

1

2

3

4

5

6

0 1 2 3 4 5

Wav

elet

Ent

ropy

GLR

vl=1.56 m/svl=1.76 m/svl=1.95 m/s

Figure 27. Wavelet entropy flow map for gas/Newtonian flow.Appl. Sci. 2020, 10, x FOR PEER REVIEW 26 of 31

Figure 28. Wavelet entropy flow map for gas/non-Newtonian flow.

The wavelet norm entropy value may vary with higher resolution sensors. Wavelet norm entropy depends on the pressure fluctuation signal; therefore, the more sensitively and precisely the sensor can detect the flow condition, the more accurate the wavelet norm entropy will be. However, the wavelet entropy change pattern with a different type of flow regime should remain similar. Fan et al. (2013) and Sun et al. (2013) [32] both observed similar wavelet entropy change patterns. However, their sensors, experimental setups, and experimental conditions were different.

4. Conclusions

In this experimental work, air/Newtonian and air/non-Newtonian fluids were evaluated in horizontal flow loop conditions. The flow maps of both systems were established, and the type of flow regimes were determined for both air/Newtonian and air/non-Newtonian fluid flow conditions. The flow regime analysis revealed that the slug flow regime condition was observed for most of the studied conditions. The slug frequency analysis illustrated that the gas/Newtonian and gas/non-Newtonian fluids have a significant difference in slug properties, mainly due to the difference in viscosity of the systems. The air/water slug frequency decreased to an air velocity of approximately 5 m/s and then increased with increased air velocity. On the contrary, the air/xanthan gum solution did not demonstrate a similar effect, with the slug frequency slowly decreasing with increased air velocity within the experimental data range. Moreover, both the air/Newtonian and air/non-Newtonian slug frequency was validated with the slug frequency correlations available in the literature, and experimental data were found to be a satisfactory analog of model predictions. Although the pressure sensor used in this study was not able to provide high-frequency and high-resolution data, the wavelet decomposition and wavelet norm entropy were nonetheless calculated, and these offered recognizable flow characteristics for bubble, bubble-elongated bubble, and slug flow patterns. In addition, 1-D wavelet packet decomposition was shown to be a useful method to identify different features of multiphase flow and for recognizing different flow patterns. Therefore, the present study can pave the way for the development of safer designs in pipe flow, especially in oil and gas pipelines, where undesired slug flow can lead to corrosion issues.

Author Contributions: M.M., M.A.R., and S.I. conceived the presented idea. M.A.R. and S.I. developed the theory, and M.M. performed the experimentations. M.A.R. and S.I. supervised the findings of this work. M.S.K.

Figure 28. Wavelet entropy flow map for gas/non-Newtonian flow.

The wavelet norm entropy value may vary with higher resolution sensors. Wavelet norm entropydepends on the pressure fluctuation signal; therefore, the more sensitively and precisely the sensor candetect the flow condition, the more accurate the wavelet norm entropy will be. However, the waveletentropy change pattern with a different type of flow regime should remain similar. Fan et al. (2013)and Sun et al. (2013) [32] both observed similar wavelet entropy change patterns. However, theirsensors, experimental setups, and experimental conditions were different.

4. Conclusions

In this experimental work, air/Newtonian and air/non-Newtonian fluids were evaluated inhorizontal flow loop conditions. The flow maps of both systems were established, and the type offlow regimes were determined for both air/Newtonian and air/non-Newtonian fluid flow conditions.The flow regime analysis revealed that the slug flow regime condition was observed for mostof the studied conditions. The slug frequency analysis illustrated that the gas/Newtonian andgas/non-Newtonian fluids have a significant difference in slug properties, mainly due to the differencein viscosity of the systems. The air/water slug frequency decreased to an air velocity of approximately5 m/s and then increased with increased air velocity. On the contrary, the air/xanthan gum solutiondid not demonstrate a similar effect, with the slug frequency slowly decreasing with increased airvelocity within the experimental data range. Moreover, both the air/Newtonian and air/non-Newtonian

Appl. Sci. 2020, 10, 3272 24 of 27

slug frequency was validated with the slug frequency correlations available in the literature, andexperimental data were found to be a satisfactory analog of model predictions. Although the pressuresensor used in this study was not able to provide high-frequency and high-resolution data, thewavelet decomposition and wavelet norm entropy were nonetheless calculated, and these offeredrecognizable flow characteristics for bubble, bubble-elongated bubble, and slug flow patterns. Inaddition, 1-D wavelet packet decomposition was shown to be a useful method to identify differentfeatures of multiphase flow and for recognizing different flow patterns. Therefore, the present studycan pave the way for the development of safer designs in pipe flow, especially in oil and gas pipelines,where undesired slug flow can lead to corrosion issues.

Author Contributions: M.M., M.A.R. and S.I. conceived the presented idea. M.A.R. and S.I. developed the theory,and M.M. performed the experimentations. M.A.R. and S.I. supervised the findings of this work. M.S.K. andM.M. wrote the manuscript, and M.A.R. and S.I. reviewed the article. Overall, all the authors contributed to thebetterment of the final manuscript. All authors have read and agreed to the published version of the manuscript.

Funding: The APC was partially funded by Qatar National Research Fund via the grant NPRP10-0101-170091.

Acknowledgments: The authors would like to acknowledge the Faculty of Engineering and Applied Scienceat the Memorial University of Newfoundland for providing laboratory facilities for this research. In addition,this publication was made possible by the grant NPRP10-0101-170091 from Qatar National Research Fund(a member of the Qatar Foundation). Statements made herein are solely the responsibility of the authors.

Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations

vsl Superficial Liquid Velocity

vl Liquid Inlet Velocityρw Water Density = 996 Kg/M3

lp Pipe Lengthvm No-Slip Mixture Velocityfs Slug Frequencyfsn Slug Frequency for Non-Newtonian FluidN fs Froude Numbervmn No-Slip Mixture Velocity for Non-Newtonian FluidN fsn Froude Number for Non-Newtonian FluidRew Water Reynolds Numberρn Non-Newtonian Density = 1002 Kg/M3

σ Shear Stressx Signal

Ψ jk(x) Wavelet Basek Wavelet Leveln Power Law Indexvs

g Superficial Gas Velocityvg Gas Inlet Velocityµw Water Viscosity 998 At 20 ◦Cd Pipe Diameterλ Liquid Volume FractionvG True Average Gas Velocity in Multiphase Flowg Acceleration due to gravity

v0m Minimum Slug Frequency in The Graph = 6 m/s

vlnNon-Newtonian Liquid Inlet Velocity

Ren Non-Newtonian Reynolds Numberγ.

yx Shear Rateµn Apparent Viscosity

W( j, k) Wavelet Transformϕ j,k(x) Scale Function

j Wavelet Scalesαg Void fraction

Appl. Sci. 2020, 10, 3272 25 of 27

References

1. Epelle, E.I.; Gerogiorgis, D.I. A review of technological advances and open challenges for oil and gas drillingsystems engineering. AIChE J. 2020, 66, e16842. [CrossRef]

2. Zahid, A.A.; Rehman, S.R.; Rushd, S.; Hasan, A.; Rahman, M.A. Experimental investigation of multiphaseflow behavior in drilling annuli using high speed visualization technique. Front. Energy 2018. [CrossRef]

3. Khan, M.S.; Bavoh, C.B.; Partoon, B.; Nashed, O.; Lal, B.; Mellon, N.B. Impacts of ammonium based ionicliquids alkyl chain on thermodynamic hydrate inhibition for carbon dioxide rich binary gas. J. Mol. Liq.2018, 261, 283–290. [CrossRef]

4. Kamyab, M.; Rasouli, V.; Cavanough, G.; Mandal, S. Challenges of cuttings transport in micro- boreholecoiled tubing drilling for mineral exploration. WIT Trans. Eng. Sci. 2013, 81, 109–120.

5. Kelin, W.; Tie, Y.; Xiaofeng, S.; Shuai, S.; Shizhu, L. Review and analysis of cuttings transport in complexstructural wells. Open Fuels Energy Sci. J. 2013, 6, 9–17. [CrossRef]

6. Qasim, A.; Khan, M.S.; Lal, B.; Shariff, A.M. A perspective on dual purpose gas hydrate and corrosioninhibitors for flow assurance. J. Pet. Sci. Eng. 2019, 183, 106418. [CrossRef]

7. Taitel, Y.; Dukler, A.E. A model for predicting flow regime transitions in horizontal and near horizontalgas-liquid flow. AIChE J. 1976, 22, 47–55. [CrossRef]

8. Shojaei, M.J.; Rodríguez de Castro, A.; Méheust, Y.; Shokri, N. Dynamics of foam flow in a rock fracture:Effects of aperture variation on apparent shear viscosity and bubble morphology. J. Colloid Interface Sci. 2019,552, 464–475. [CrossRef]

9. Mandhane, J.M.; Gregory, G.A.; Aziz, K. A flow pattern map for gas-liquid flow in horizontal pipes. Int. J.Multiph. Flow 1974, 1, 537–553. [CrossRef]

10. Chhabra, R.P.; Richardson, J.F. Prediction of flow pattern for the co-current flow of gas and non-newtonianliquid in horizontal pipes. Can. J. Chem. Eng. 1984, 62, 449–454. [CrossRef]

11. Azarinezhad-Mohammadi, R. A Chemical Based Wet Cold Flow Approach for Addressing Hydrate FlowAssurance Problems. Ph.D. Thesis, Heriot-Watt University , Edinburgh, UK, 2010.

12. Joshi, S.V.; Grasso, G.A.; Lafond, P.G.; Rao, I.; Webb, E.; Zerpa, L.E.; Sloan, E.D.; Koh, C.A.; Sum, A.K.Experimental flowloop investigations of gas hydrate formation in high water cut systems. Chem. Eng. Sci.2013, 97, 198–209. [CrossRef]

13. Khan, M.S.; Lal, B.; Bavoh, C.B.; Keong, L.K.; Bustam, A. Influence of ammonium based compounds for gashydrate mitigation: A short review. Indian J. Sci. Technol. 2017, 10, 1–6. [CrossRef]

14. Al-Safran, E. Investigation and prediction of slug frequency in gas/liquid horizontal pipe flow. J. Pet. Sci. Eng.2009, 69, 143–155. [CrossRef]

15. Sultan, R.A.; Alfarek, S.; Rahman, M.A.; Zendehboudi, S. CFD and experimental approach on three phasegas-liquid-solid Newtonian fluid flow in horizontal pipes. Int. J. Comput. Methods Exp. Meas. 2018, 7, 33–44.[CrossRef]

16. Zadrazil, I.; Markides, C.N. An experimental characterization of liquid films in downwards co-currentgas-liquid annular flow by particle image and tracking velocimetry. Int. J. Multiph. Flow 2014, 67, 42–53.[CrossRef]

17. Cerqueira, R.F.L.; Paladino, E.E.; Ynumaru, B.K.; Maliska, C.R. Image processing techniques for themeasurement of two-phase bubbly pipe flows using particle image and tracking velocimetry (PIV/PTV).Chem. Eng. Sci. 2018, 189, 1–23. [CrossRef]

18. Talley, J.D.; Worosz, T.; Kim, S.; Buchanan, J.R. Characterization of horizontal air–water two-phase flow in around pipe part I: Flow visualization. Int. J. Multiph. Flow 2015, 76, 212–222. [CrossRef]

19. Majumder, S.K.; Ghosh, S.; Kundu, G.; Mitra, A.K. Frictional pressure drop of gas-Newtonian and gas-nonNewtonian slug flow in vertical pipe. Int. J. Chem. React. Eng. 2011. [CrossRef]

20. Ehinmowo, A.B.; Cao, Y.; Yeung, H.C. Slug flow in large diameter pipeline—Riser systems: Prediction andmitigation. Covenant J. Eng. Technol. 2018, 2, 1–21.

21. Majumder, S.K.; Ghosh, S.; Mitra, A.K.; Kundu, G. Gas-Newtonian and gas-non-Newtonian slug flow invertical pipe, part I: Gas holdup characteristics. Int. J. Chem. React. Eng. 2010. [CrossRef]

22. Abed, E.M.; Ghoben, K. Gas-liquid slug frequency and slug unit length in horizontal pipes. Iraqi J. Mech.Mater. Eng. 2015, 15, 166–180.

Appl. Sci. 2020, 10, 3272 26 of 27

23. Zabaras, G.J.; Shell, E.; Co, P.T. Prediction of Slug Frequency for Gas/Liquid Flows; Society of Petroleum Engineers:Richardson, TX, USA, 1999.

24. Sultan, R.A.; Rahman, M.A.; Rushd, S.; Zendehboudi, S.; Kelessidis, V.C. CFD analysis of pressure losses anddeposition velocities in horizontal annuli. Int. J. Chem. Eng. 2019, 2019, 1–17. [CrossRef]

25. Daza Gómez, M.A.; Ratkovich, N.; Pereyra, E.; Poesio, P. CFD simulations of two-phase gas/non-Newtonianfluid flows in pipes. In Proceedings of the 3rd World Congress on Momentum, Heat and Mass Transfer(MHMT’18), Budapest, Hungary, 12–14 April 2018.

26. Dziubinski, M.; Fidos, H.; Sosno, M. The flow pattern map of a two-phase non-Newtonian liquid-gas flow inthe vertical pipe. Int. J. Multiph. Flow 2004, 30, 551–563. [CrossRef]

27. Katzbauer, B. Properties and applications of xanthan gum. Polym. Degrad. Stab. 1998, 59, 81–84. [CrossRef]28. Bavoh, C.B.; Md Yuha, Y.B.; Tay, W.H.; Ofei, T.N.; Lal, B.; Mukhtar, H. Experimental and modelling of the

impact of quaternary ammonium salts/ionic liquid on the rheological and hydrate inhibition properties ofxanthan gum water-based muds for drilling gas hydrate-bearing rocks. J. Pet. Sci. Eng. 2019, 183, 106468.[CrossRef]

29. Owusu, E.B.; Tsegab, H.; Sum, C.W.; Padmanabhan, E.; Qasim, A.; Khan, M.S.; Lal, B.; Ismail, M.C.; Rostani, K.;Shojaei, M.J.; et al. Synthesis, characterization, stability and thermal conductivity of multi-walled carbonnanotubes (MWCNTs) and eco-friendly jatropha seed oil based nanofluid: An experimental investigationand modeling approach. J. Mol. Liq. 2019, 293, 6–14.

30. Tutu, N.K. Pressure fluctuations and flow pattern recognition in vertical two phase gas-liquid flows. Int. J.Multiph. Flow 1982, 8, 443–447. [CrossRef]

31. Drahoš, J.; Cermák, J.; Selucky, K.; Ebner, L. Characterization of hydrodynamic regimes in horizontaltwo-phase flow: Part II: Analysis of wall pressure fluctuations. Chem. Eng. Process. Process Intensif. 1987, 22,45–52. [CrossRef]

32. Sun, Z.; Shao, S.; Gong, H. Gas-liquid flow pattern recognition based on wavelet packet energy entropy ofvortex-induced pressure fluctuation. Meas. Sci. Rev. 2013, 13, 83–88. [CrossRef]

33. Blaney, S. Gamma Radiation Methods for Clamp-On Multiphase Flow Metering; Cranfield University: Bedford,UK, 2008.

34. Park, S.H.; Kim, S.D. Characterization of pressure signals in a bubble column by wavelet packet transform.Korean J. Chem. Eng. 2003, 20, 128–132. [CrossRef]

35. De Fang, L.; Liu, R.; Lu, Q.H.; Wang, X.J.; Liang, Y.J. The flow pattern transition identification and interphasesforce detection of gas-liquid two-phase flow. AASRI Procedia 2012, 3, 534–539. [CrossRef]

36. Elperin, T.; Klochko, M. Flow regime identification in a two-phase flow using wavelet transform. Exp. Fluids2002, 32, 674–682. [CrossRef]

37. Kouba, G.E.; Jepson, W.P. The flow of slugs in horizontal, two-phase pipelines. J. Energy Resour. Technol.1990, 112, 20–24. [CrossRef]

38. Hubbard, M.G. An Analysis of Horizontal Gas-Liquid Slug Flow; University of Houston: Houston, TX, USA, 1965.39. Hubbard, M.G.; Dukler, A.E. The characterization of flow regimes for horizontal two-phase flow. Proc. Heat

Transf. Fluid 1966, 1996, 100–121.40. Otten, L.; Fayed, A.S. Slug velocity and slug frequency measurements in concurrent air-non-Newtonian slug

flow. Trans. Inst. Chem. Eng. 1977, 55, 64–67.41. Gregory, G.A.; Scott, D.S. Correlation of liquid slug velocity and frequency in horizontal concurrent gas-liquid

slug flow. AIChE J. 1969, 15, 933–935. [CrossRef]42. Rosehart, R.G.; Rhodes, E.; Scott, D.S. Studies of gas liquid (non-Newtonian) slug flow: Void fraction meter,

void fraction and slug characteristics. Chem. Eng. J. 1975, 10, 57–64. [CrossRef]43. Greskovich, E.J.; Shrier, A.L. Slug frequency in horizontal gas-liquid slug flow. Ind. Eng. Chem. Process

Des. Dev. 1972, 11, 317–318. [CrossRef]44. Zabaras, G.J. Prediction of slug frequency for gas-liquid flows. In Proceedings of the SPE Annual Technical

Conference and Exhibition, Houston, TX, USA, 3–6 October 1999.45. Heywood, N.I.; Richardson, J.F. Slug flow of air water mixtures in a horizontal pipe: Determination of liquid

holdup by γ-ray absorption. Chem. Eng. Sci. 1979, 34, 17–30. [CrossRef]46. Shea, R.; Eidsmoen, H.; Nordsveen, M.; Rasmussen, J.; Xu, Z.; Nossen, J. Slug frequency prediction method

comparison. In Proceedings of the 4th North American Conference on Multiphase Technology, Banff, AB,Canada, 3–4 June 2004; pp. 227–237.

Appl. Sci. 2020, 10, 3272 27 of 27

47. Picchi, D.; Manerba, Y.; Correra, S.; Margarone, M.; Poesio, P. Gas/shear-thinning liquid flows through pipes:Modeling and experiments. Int. J. Multiph. Flow 2015, 73, 217–226. [CrossRef]

48. Misiti, M.; Misiti, Y.; Oppenheim, G.; Poggi, J.-M. Wavelet Toolbox; MathWorks Inc.: Natick, MA, USA, 1996;Volume 15, p. 21.

49. Gao, R.X.; Yan, R. Wavelets: Theory and Applications for Manufacturing; Springer Science & Business Media:Berlin/Heidelberg, Germany, 2010.

50. Addison, P.S. The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science,Engineering, Medicine and Finance; CRC Press: Boca Raton, FL, USA, 2017.

51. Walker, J.S. A Primer on Wavelets and Their Scientific Applications; CRC Press: Boca Raton, FL, USA, 2008.52. Coifman, R.R.; Wickerhauser, M.V. Entropy-based algorithms for best basis selection. IEEE Trans. Inf. Theory

1992, 38, 713–718. [CrossRef]53. Uyar, M.; Yildirim, S.; Gencoglu, M.T. An effective wavelet-based feature extraction method for classification

of power quality disturbance signals. Electr. Power Syst. Res. 2008, 78, 1747–1755. [CrossRef]54. Fan, C.; Ding, Y.; Ren, X. Wavelet entropy applied in gas-liquid two-phase flow. In Proceedings of the 32nd

Chinese Control Conference (CCC), Xi’an, China, 26–28 July 2013; pp. 8623–8627.55. Yu, Y.; Junsheng, C. A roller bearing fault diagnosis method based on EMD energy entropy and ANN.

J. Sound Vib. 2006, 294, 269–277. [CrossRef]56. Chhabra, R.P.; Richardson, J.F. Non-Newtonian Flow: Fundamentals and Engineering Applications;

Butterworth-Heinemann: Oxford, UK, 1999.57. Shaikh, A.; Al-Dahhan, M.H. A review on flow regime transition in bubble columns. Int. J. Chem. React. Eng. 2007.

[CrossRef]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (http://creativecommons.org/licenses/by/4.0/).